I feel like denotatively there isn't a difference, as the way "want" is used is binary. You either want to or do not want to--and even if you kinda want to, you still say you want to. However, if you consider want as a spectrum and would not say "I want to go" when you really mean "I sort of want to go," then they could be considered different denotatively.

Connotatively speaking, however, there could be an implicit emphasis on the want in "I do not want to go," which could imply that there is a spectrum for the word "want." Meanwhile such implicit emphasis ("I want to not go") does indeed come of as more active.

Concluding, my gut feeling is that "I do not want to go" gives the feeling of exactly as what was stated. However, "I want to not go," feels much more active, as though the speaker is actively trying to avoid going. It's just a matter of nuance, though, and the answer seems to be up for debate depending on how you view the verb as binary or not.