I've never used the dialog boxes for Mixed (for this exact reason--it's too hard to tell what the heck they're doing).
So are you basically asking about the difference between using a /Random subcommand and a /Repeated? You can have a |Subjects option in both.
They're doing totally different things, though. The Repeated statement is adjusting the residuals to take into account that they're correlated for the same Subject. (And you can choose the exact pattern of correlations by specifying the Covariance matrix).
The Random statement is actually creating a new parameter in the model to take the effect of Subject into account. In a simple one-way repeated measures model (with or without a between subjects factor), you can actually get the exact same result either way, if you specify the covariance matrices right. But if your model gets complicated, you have to be more careful.
If you describe your model more specifically (and it's not too complicated), I can try to help you specify it. This gets pretty tricky.