Difference between SUR and Simultaneous Equation Model Seemingly Unrelated Regression (SUR), and Simultaneous Equation Model (SEM) sound very similar to me. What is the difference between them? 
 A: SUR and SEM
Loosely speaking, Seemingly Unrelated Regression (SUR) is a method for estimating the parameters in a system of equations. In comparison, a Simultaneous Equation Model (SEM) is a collection of equations - not an estimation method. In other words, the former refers to a method while the latter refers to a class of model. 
SEM Parameters
It's worth noting that the parameters in a SEM can be estimated or calibrated and the equations themselves can be stochastic or deterministic. In constructing a SEM, the estimation methods can vary; for example, single-equation or system-equation methods (least squares (1SLS), seemingly unrelated regression (SUR), full information maximum likelihood (FIML), etc.) can be used. In practice, each method has advantages and disadvantages.
SEM Solution
Once the parameters of a SEM have been decided (by estimation or calibration), the next step is to solve the model - for values of the endogenous variables. One popular algorithm for doing this is the Gauss-Seidel method. 
When the model is solved for a baseline or control solution, various techniques can be used to assess the model's properties; mostly done through simulation. For example, simulating the model in-sample can be used to evaluate the tracking or forecast performance of the SEM while simulations out-of-sample can reveal the (in)stability of the model. 
Scenario (shock) analysis is also possible by making perturbations to the model (exogenous variables, parameter values, etc.) and making comparisons to a baseline solution.
Edit: Finally, let me stress that SUR could be used to estimate the parameters that appear in a SEM, but SUR could not be used to solve a SEM. SUR is not a solution method - unlike the Gauss-Seidel algorithm which is.
