They both mean "very small," but teeny" is more of an informal child's word. With children, we often say "teeny-tiny" to emphasize that something is really, really small (notice that I just doubled the word "really."). Sometimes, in informal speech, by doubling a word or pairing two words that mean the same thing, we are adding emphasis. If I say that from the airplane people look like "teeny tiny" bugs, that's smaller than just saying "tiny" and it adds emphasis to the smallness of how the people appear. We'd be much more likely to use a phrase like this with children than with adults.

"Teeny" is a cutesy version of "tiny". You would never use it in standard speech. However, you can use it in song! :D http://www.youtube.com/watch?v=ICkWjdQuK7Q