Using "not very" is plain speech. Normally choose this.
Using "none too" is a rather rhetorical device. It is much less common than "not very", though it is perfectly correct and idiomatic. It's relative rarity makes it "marked" - a speaker using "none too" is making a choice not to use the plain form, for its effect, not for its meaning.
There is a sense of irony. Saying "I'm none too happy with it", suggests that I'm actually rather unhappy about it, but I'm making a dramatic understatement. There are subtleties here. This technique is called by its Greek name - litotes - in classical rhetoric.
But "not very" can have a similar effect. It is a more common form and the rhetorical effect is lessened