Partial vs. general equilibrium Microeconomic models are usually classified as partial and general equilibrium models. As a layman, I understand that partial equilibrium focuses attention on a few economic variables to find the equilibrium, while general eq. models capture a larger interaction.

Besides this, what are the key differences between the two kinds of modelling? Are there advantages and downsides to each of them?
Let's put the succinct answer by @TheAlmightyBob into an abstract model:

We want to model the labor market.

Markets' structure assumptions: goods market and labor markets are perfectly competitive. All participants are "too small" economically, and they cannot affect equilibrium price through their quantities demanded/supplied - they are "price takers". Markets "clear" - i.e. prices adjust so that quantity actually supplied equals quantity actually bought.

Agents assumption: There are $n$ identical workers, and $m$ identical firms, that participate in the market. Both populations are fixed.

Other assumptions: a) deterministic environment, b) one perishable good produced, c) model in "real terms" (real wage etc, scaled by the price of the good produced).

The typical firm produces according to the technology
$$Y_j = F_j(K_j,L_j;\mathbf q)  \tag{1}$$

where $\mathbf q$ is a vector of parameters. Perfect competition in the goods market, and a perishable good imply that all output produced is sold.The goal of the firm is maximization of capital returns over the choice of labor.  

$$\max_{L_j} \pi_j = F_j(K_j,L_j;\mathbf q) - wL_j$$

We are modelling the labor market, so we are interested in the first-order condition

$$\frac {\partial \pi_j}{\partial L_j} = 0 \tag{2}$$
and the corresponding input demand schedule

$$L_j^* = L_j^*\left(K_j, \mathbf q, w\right) \tag{3}$$

Total Labor demand is $L_d = m\cdot L_j^*$. 
The labor market equilibrium assumption implies

$$ L_d = L_s \Rightarrow m\cdot L_j^*\left(K_j, \mathbf q, w\right) = L_s \tag{4}$$

which implicitly expresses the equilibrium wage as a function of technology constants, of per-firm capital, and of labor supplied. In order to fully characterize the labor market, we need to derive also the optimal labor supply.  

Each identical worker derives utility from consumption and leisure, subject to a biological limit of available time, $T$, and the budget constraint that consumption equals wage income:

$$\max_{L_i} U(C_i, T-L_i;\mathbf \gamma),\;\; \text{s.t.} \;C_i= wL_i$$

where $\gamma$ is a vector of preference parameters, indicating the relative weight between utility from consumption, and from leisure.
This will give us individual labor supply as

$$L_i^* = L_i^*(T,w, \mathbf \gamma)  \tag{5}$$

and total labor supply is $L_s = n\cdot L_i^*$. Plugging this into $(4)$  we obtain

$$mL_j^*\left(K_j, \mathbf q, w\right) =n L_i^*(T,w, \mathbf \gamma) \tag{6}$$

If we stop here, we have a partial equilibrium model that examines the labor market. We have fully described the market, and the goals and the constraints of the participants in it (firms and workers), related to the specific market. We can perform comparative statics in order to see how the various components of $(6)$ affect the equilibrium wage. Among them, there is the capital-per-firm term, whose effects on wage we can also consider based on $(6)$, by treating it as varying arbitrarily. 

In order to turn this model into a general equilibrium model:
a) We need to specify things about capital: who owns it/controls it/makes decisions on it. What is the objective functions of these decision makers. This will lead us to an optimal $K_j^*$ as a function of the structure we will impose here. Then, comparative statics with respect to $K_j$ will turn into comparative statics with respect to the factors that affect the determination of $K_j^*$, which may very well prove to involve also $\mathbf q, w$ and even the other parameters in $(6)$, changing in this way the comparative statics results obtained in a partial equilibrium setting.  

b) We also need to take into account any macroeconomic identities that characterize this economy, something along the lines of $mY_j \equiv ...$ where the right hand side will be determined by the assumptions we make related to capital, but also, for example, by whether we will assume that the economy is closed or open, or partially open to the outside economic system.

So, apart from being more complicated as a model, it may also lead us to different conclusions than partial equilibrium analysis.