In design of experiments, these two are called the fixed effects model and the random effects model.

Most tests in regressoin are the same, but NOT all are.

It really comes down to whether or not you're tying to make an inference about all x's and in that case you're randomly picking a subset of those x's. In the fixed case, you selct which x's you want BUT you cannot make any inferences about the x's you did not select, because you didn't randomly pick the x-values.