Seemingly Unrelated Regression (SUR), and Simultaneous Equation Model (SEM) sound very similar to me. What is the difference between them?
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.
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.
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.