One might add to Kuba's answer that x and x$ are distinct symbols. The $ does not mean anything. It is added to the symbol name of x to create a different symbol. Both fortunately and unfortunately, this behavior is predictable. If you wanted to inject a function body e that has the Function variable in it, you can use x$ instead of x and count on the variable being rewritten x$.
With[{e = x$}, Function[x, e]]
(* Function[x$, x$] *)
On the other hand, if you wanted the symbol x$ to be distinct symbol from the Function variable, then the above code wouldn't work. You would have to use the ReplaceAll form Function[x, e] /. e -> x$. But since this rewriting happens in various contexts, better advice is the following:
Do not end the names of variables with a $.
