Here, make is used in a context of "to cause something":
You make me cry — meaning, I was not crying, but because of you I start.
Become is simply "to change {one's own} state".
Imagine Alice sees a sad Bob. Alice tells a funny joke to Bob, and he becomes happy. The following sentences are all valid:
- Alice made Bob happy;
- Bob became happy;
- Alice made Bob
to (1) become happy;
So, as you see, to make applies to a causing subject, while to become applies to an object that changes.
(1) Thanks to @StoneyB; make takes bare infinitive, so can't use to here.