Josh Regev's answer explains the difference very well. But a a baseball analogy occurs to me that might clarify the meaning even further:
Carl ran to second base.
Carl ran for second base.
The first sentence means that Carl succeeded in getting to second base. The second sentence means that Carl attempted to run to second base, but doesn't state whether he succeeded in getting there or not.
Now, this is clear in past tense, but in present progressive tense, the meanings run together more. Consider this:
Carl is running to second base.
Carl is running for second base.
Since the tense means that he is in the act of running to — or for — second base, he hasn't gotten there yet in either case. So we don't know yet whether he is going to make it or not:
And Yastrzemski is running to second! There's the throw ... and he is .... OUT at second on a great throw from Parker! Yastrzemski ran for second and he didn't make it.
In that last sentence, you wouldn't use to, because the action is over and done with. Yaz tried, but he never made it to second, so he ran for second.