What is the appeal of DSGE models? As far as I understand people like them, because they fit the data quite well when compared to VAR's for example while also having an economic interpretation.
They appear to explain observations in a more general way than VAR's.

However for me personally that point falls flat as I can not convince myself to see the assumptions (everything is differentiable instead of discrete, goods don't depend on each other for production, people have no skills, etc.) to be a sensible simplification of reality. So I'm left wondering if DSGE models fit the data despite their assumptions. After all they have quite a number of parameters, which might make them quite flexible.

Or do DSGE models fit the data better than VAR's or other models like neural networks?

Did people investigate the "expressiveness" of DSGE models? With expressiveness I mean the set of functions(DSGE models predict inflation as a function of time for example) you can approximate by choosing the parameters of a given DSGE model.
To summarize what I wrote below, there seem to be at least two points:


In the Smets and Wouters (ECB) research, which is apparently responsible for a lot of the fame of DSGE, they indeed found it superior in forecasting accuracy to (B)VARs, especially long-term. On the other hand, other research (including some by other ECB researchers) doesn't seem to be so uniformly conclusive in this (accuracy) regard. Alas, it doesn't look like there is anything resembling a systematic review on this issue; perhaps the variations in data sets and methodology preclude one.
In the more academic realm, proponents of DSGE (e.g. Christiano) argue that DSGE has deep explanatory powers for some inflation-related lags, making them easy to understand.




I'm sure someone will give you a more technical answer, but basically DSGE works well for monetary policy (inflation targeting) etc. 


  Part of the recent popularity of DSGE models is due to work by Smets and
  Wouters (2003), who document that a modified version of a New Keynesian model
  developed by Christiano, Eichenbaum, and Evans (2005) is able to track and forecast
  euro area time series as well as, if not better than, a vector autoregression (VAR) estimated with Bayesian techniques. 


In other areas like fiscal policy DSGE hasn't seen that much success/application, insofar anyway.

However, the New Keynesian DSGEs have their critics, particularly in the Minneapolis Fed; see Chari, Kehoe, and McGrattan.


  Proponents of the New Keynesian model argue that it is promising for two reasons. It
  represents a detailed economy that can generate the type of wedges we see in the data from
  interpretable primitive shocks; and second, it has enough microfoundations that both their
  shocks and parameters are structural, in that they can reasonably be argued to be invariant
  to monetary policy shocks. A model with both of these features would potentially be useful
  for monetary policy analysis.
  
  We disagree. We argue that these models cannot generate the type of wedges we see
  in the data from interpretable primitive shocks. And it is doubtful that many of the features
  added on in the quantitative implementation of the models are structural. Hence, the models
  are not yet useful for policy analysis.


(The first paper I quoted from is from the Atlanta Fed.)

Also, VAR and DSGE are not orthogonal or alternative as you suggest; from the Atlanta paper:


  It has long been recognized (for example, Sims 1980) that a tight relationship exists between dynamic
  equilibrium models and VARs. Imagine the following thought experiment, where for
  the moment the vector of DSGE model parameters is fixed. We generate 1 million
  observations from the DSGE model—that is, we generate a sequence of shocks
  (monetary policy, technology, etc.), feed them trough the DSGE model, and obtain
  artificial data. Next, we estimate a VAR with p lags on these artificial data. If the
  DSGE model is covariance stationary, then the estimated VAR provides an approximation to the DSGE model with the property that its first p autocovariances are
  equivalent to the first p autocovariances of the DSGE model. By including more and
  more lags we can in principle match more and more autocovariances and increase the
  accuracy of the VAR approximation of the DSGE model. Now imagine that the data
  generation is repeated using different parameter values for the DSGE model. As long
  as the DSGE model parameter space is small compared to the VAR parameter space,
  a restriction function can be traced that maps the DSGE parameters into a VAR
  parameter subspace. Hence, estimating a DSGE model is (almost) like estimating a
  VAR with cross-equation restrictions. 


That paper is somewhat limited (empirically) in that it looks at a single (New Keynesian) DSGE and VAR-izes it.

For a more general comparison see: Raffaella Giacomini, The Relationship Between DSGE and VAR Models. It does mention some caveats e.g.


  a meaningful discussion of the relationship between DSGE and VAR
  models can only be carried out in the context of log-linearized DSGE models, taking for granted
  the adequacy of the linear approximation and ignoring possible nonlinear dynamics that cannot
  be replicated by linear VAR models.


It goes on to point out that non-linear DSGEs are (still) a minority in current research, but that that sub-field appears to be growing pretty fast.

The claim to DSGE empirical superiority (as estimated by the more widespread Bayesian techniques, rather than by embedding in VARs) is probably in the narrow(er) context of specific papers; the 2007-updated Smets and Wouter does indeed make such claims. In the short term, they only find a a difference for unconstrained VARs and they find that a BVAR model does as well as their DSGE; but for a longer forecast they indeed found their DSGE superior to both VAR/BVAR.


  In this Section we compare the out-of-sample forecast performance of the estimated DSGE model with
  that of various VARs estimated on the same data set. The marginal likelihood, which can be interpreted as
  a summary statistic for the model’s out-of-sample prediction performance, forms a natural benchmark for
  comparing the DSGE model with alternative specifications and other statistical models. However, a Sims (2003) has pointed out it is important to use a training sample in order to standardize the prior
  distribution across widely different models. In order to check for robustness, we also consider a more
  traditional out-of-sample RMSE forecast exercise in this section. 
  
  
  
  Table 2 compares the marginal likelihood of the DSGE model and various unconstrained VAR models,
  all estimated over the full sample period (1966:1 – 2004:4) and using the period 1956:1 – 1965:4 as a
  training sample. Several results are worth emphasizing. First, the tightly parameterized DSGE model
  performs much better than an unconstrained VAR in the same vector of observable variables, Υt (first
  column of Table 2). The bad empirical performance of unconstrained VARs may not be too surprising, as
  it is known that over-parameterized models typically perform poorly in out-of-sample forecast exercises.
  One indication of this is that the marginal likelihood of the unconstrained VAR model deteriorates
  quickly as the lag order increases. For that reason, in the second column of Table 2, we consider the
  Bayesian VAR model proposed by Sims and Zha (1998). This BVAR combines a Minnesota-type prior
  (see Litterman, 1984) with priors that take into account the degree of persistence and cointegration in the
  variables. In order to allow the data to decide on the degree of persistence and cointegration, in this
  BVAR we enter real GDP, consumption, investment and the real wage in log levels. When setting the
  tightness of the prior, we choose a set of parameters recommended by Sims (2003) for quarterly data.
  The second column of Table 2 shows that the marginal likelihood of the Sims-Zha BVAR increases
  significantly compared to the unconstrained VAR. Moreover, the best BVAR model (BVAR(4)) does as
  well as the DSGE model.
  
  Overall, the comparison of marginal likelihoods shows that the estimated DSGE model can compete with
  standard BVAR models in terms of empirical one-step-ahead prediction performance. These results are
  confirmed by a more traditional out-of sample forecasting exercise reported in Table 3. Table 3 reports
  out-of-sample RMSEs for different forecast horizons over the period 1990:1 to 2004:4. For this exercise the VAR(1), BVAR(4) and DSGE model were initially estimated over the sample 1966:1 - 1989:4. The
  models were then used to forecast the seven data series contained in Υt from 1990:1 to 2004:4, whereby
  the VAR(1) and BVAR(4) models were re-estimated every quarter, whereas the DSGE model was reestimated every year. The measure of overall performance reported in the last column of Table 3 is the
  log determinant of the uncentered forecast error covariance matrix. 
  
  
  
  The out-of-sample forecast statistics confirm the good forecast performance of the DSGE model relative
  to the VAR and BVAR models. At the one-quarter ahead horizon, the BVAR(4) and the DSGE model
  improve with about the same magnitude over the VAR(1) model, confirming the results from Table 2.
  However, over longer horizons up to three years, the DSGE model does considerably better than both the
  VAR(1) and BVAR(4) model. Somewhat surprisingly, the BVAR(4) model performs worse than the
  simple VAR(1) model at longer horizons. Moreover, the improvement appears to be quite uniform across
  the seven macro variables.


However, there's somewhat newer (2009) ECB paper by different authors (Kolasa, Rubaszek, and Skrzypczyński) with less sweeping conclusions:


  In the case of ination, the DSGE and DSGE-VAR
  models with the maximum lag set to 3 and 4 are characterized by the lowest RMSFEs. The SPF [Survey of Professional Forecasters]
  and BVAR(4) are insignicantly less accurate, while the BVAR and low-order DSGE-VAR models are
  found to be the worst. Finally, the RMSFEs for interest rate forecasts formulated by all methods are
  comparable.


(There's a 2012 peer-reviewed version of that paper, if you care about that. And As an aside, somewhat disappointingly, adding (corporate) financial market frictions to their DSGE model did not uniformly improve forecasts, but adding housing market frictions did help especially with crises.)

Another paper which focuses on hybrid DSGE models (Paccagnini, 2011) has this introductory note:


  from an econometric point of view, the performance of a DSGE model is often
  tested against an estimated Vector Autoregressive model (VAR). This procedure requires a Data
  Generating Process (DGP) that is consistent with the theoretical economic model and has a finite-order VAR representation. However, the statistical representation of a DSGE model is an exact
  VAR only when all the endogenous variables are observable; otherwise, a more complex Vector
  Autoregressive Moving Average model (VARMA) is needed. As far as the VARMA representation
  is concerned, several papers (see Cooley and Dwyer (1998), Chari, Kehoe and McGrattan (2005),
  Christiano, Eichenbaum and Vigfusson (2006), Ravenna (2007) and Fernandez-Villaverde, Rubio-Ramirez, Sargent and Watson (2007)) have discussed the conditions required to find an infinite-
  order VAR representation and a infinite-order VAR truncation in general terms. Moreover, the VAR is densely parameterized and appears to be misspecified and a Bayesian approach is preferred.


And the results of this paper are not amazing for DSGE:


  Two forecasting exercises have been implemented on US quarterly economy data: the first one
  considering a simple forecasting one-step-ahead strategy with different samples, and the second a
  h-steps ahead evaluation with a rolling forecasting estimation. In the first exercise, the last part of
  the sample from 2007 onwards, which is the Great Contraction period, was not taken into account.
  The main results show that the hybrid models, such as the DSGE-VAR and the DSGE-FAVAR give
  the most accurate forecasts for smaller forecasting samples. Instead, a FAVAR model outperforms
  the other models when we consider a long forecasting sample. The second forecasting experiment
  uses a rolling sample with the Great Contraction period included. The best forecasting performance
  is produced by VAR, BVAR and FAVAR, with the exception of real GDP where for h-steps ahead
  larger than 2 the DSGE-FAVAR outperforms the alternative models.
  
  
  
  


As a side-note to that last paper, FAVAR was apparently favored/suggested by Bernanke.



These (not always clear) accuracy contests aside, DSGE advocates claim that it supposedly has deep explanatory power for inflation: 


  The new monetary DSGE models are of interest not just because they represent laboratories for the analysis of important monetary policy questions. They are also of interest because
  they appear to resolve a classic empirical puzzle about the effects of monetary policy. It has
  long been thought that it is virtually impossible to explain the very slow response of inflation
  to a monetary disturbance without appealing to completely implausible assumptions about
  price frictions (see, e.g., Mankiw (2000)). However, it turns out that modern DSGE models
  do provide an account of the inertia in inflation and the strong response of real variables to
  monetary policy disturbances, without appealing to seemingly implausible parameter values.
  Moreover, the models simultaneously explain the dynamic response of the economy to other shocks. We review these important findings. We explain in detail the contribution of each
  feature of the consensus medium-sized New Keynesian model in achieving this result.
  
  The econometric technique that is particularly suited to the shock-based analysis described in the previous paragraph, is the one that matches impulse response functions estimated by vector autoregressions (VARs) with the corresponding objects in a model. Using
  US macroeconomic data, we show how the parameters of the consensus DSGE model are
  estimated by this impulse-response matching procedure. The advantage of this econometric
  approach is transparency and focus. The transparency reflects that the estimation strategy
  has a simple graphical representation, involving objects - impulse response functions - about
  which economists have strong intuition. The advantage of focus comes from the possibility of studying the empirical properties of a model without having to specify a full set of
  shocks.