The second way is slightly more emphatic. Otherwise, there is no difference.

Be in no position to do smth is an idiom meaning no right to do something. Not in a position to do smth means cannot do something in that situation, e.g. you are busy, you are sick, or you have no right

"... not in a position to ..." is grammatically more correct than " ... in no position to ..." and consequently you might see the first construction more often in text (perhaps in a business email or news story) and the second construction more often in speech (perhaps in an argument between familiars or in a politician's rhetoric). Usage of either of them adds length and ambiguity to a sentence unnecessarily.

I agree with FDMaxey. Basically, "I'm in no position to do something" implies "...at all!!" This is where the emphasis comes from. It has a sense of absoluteness. "I'm not in a position to do something" gives an example (= the article "a"), which suggests the idea that if you were in a different position, you could do something.

Thank you, Kevin Brown. Your comment has clarified the matter completely.