"I gonna take some sleep" is not in any way a proper sentence. No native speaker would ever say that. To answer your question: the difference is the first one is correct and the second one is not correct.

"I'm gonna take some sleep." is not a standard way of speaking. I can imagine it being said - but only as a punchline to a very specific joke. Stick with "I'm going to sleep" or "I'm going to go to sleep."

I've never really heard anyone say "I'm gonna take some sleep", but to me the meaning's similar to "I'm going to sleep" - "I'm gonna get some sleep" just sounds a bit more casual!

It seems that the second sentence is more likely to be used after staying up very late or getting very tired. Let's see what others have to say.