Fixed effects, first differences interpretation I am working with a fixed effects model to estimate the effect of enterprise zones on unemployment claims. 

(1)  $log(uclms_{it}) = \theta_t + \beta_1 ez_{it} + c_i + \epsilon_{it} $ where $\theta_t$ are time specific effects and $c_i$ are city specific effects. 

Another model is estimated that allows each city to have its own time trend: 
$log(uclms_{it}) = \gamma_i t + \beta_1 ez_{it} + c_i + \epsilon_{it} $ and this model is then first-differenced to obtain

(3)  $\Delta log(uclms_{it}) = \gamma_i + \beta_1 \Delta ez_{it} + \Delta \epsilon_{it} $

I was asked to estimate (1) and (3) by fixed effects and comment on possible reasons for differences in the estimate of $\beta_1$

The estimates I got were; 
from model (1) $\beta_1 = -0.1044$ with a t-ratio of -1.43 and p-value of 0.166.
from model (3) $\beta_1 = -0.252$ with a t-ratio of -2.41 and a p-value of 0.025. 

So the estimate is relatively small in model (1) compared to model (3) and is insignificant in model (1), but significant in model (3) at the 5% level. 

Am I right in thinking that, when (1) is estimated by FE, the city specific effects drop out, but the time specific effects remain. Whereas in (3) we only have city specific effects which drop out when estimated by FE. So the observed differences in $\beta_1$ arise because (1) includes time effects, but (3) does not. I'm also a bit confused about using first-differenced data in a fixed-effects model. Why would that be useful?
The biggest difference between (1) and (3) is that (3) has incidental linear trends while (1) has common time effects. Differencing (1) gives

(1a) $\Delta \log(uclms_{it}) = (\theta_t - \theta_{t-1}) + \beta_1 \Delta ez_{it} + \Delta \epsilon_{it}$. 

Comparing this to (3) you see that (3) has $\gamma_i$ in place of $\theta_t - \theta_{t-1}$ in (1a). So three things are different: (A) As you said, it is FE vs FD; (B) the trend is free (using dummies) in (1), while linear in (3); (C) the trend is common in (1) but is individual-specific in (3).

It would be interesting how much difference is introduced by each of these three points.

About why FD is useful, well it is simple and informs us about functional relationship among variables in differences. So I'd say why not.