No that's not the difference, though it is subtle. Consequence is broader, it covers punishment (a punishment is a kind of consequence) but not all consequences are punishments.
A punishment is something which is intended to link a bad feeling/outcome with the behaviour, it's negative reinforcement. You hear it a lot saying something like "Well I put him in his room but he didn't care, he just played with his toys!" or "I took the DS off him but he didn't mind because he played with other things instead" - this is implying a punishment kind of mindset, ie, it should "hurt" a little, the child should mind. Whereas with UP type of consequences you always do it for a different reason, not so that they learn to associate the "bad" behaviour with something unpleasant for them but so that it prevents the situation happening or helps to rectify it or teaches them something about how to manage that feeling/situation or why it's not a good thing to do. So while you might put them in their room or remove their DS it would be for a reason, e.g. the DS was distracting them when they were meant to be getting ready (and you'd then let them have it when they're not getting ready) or they need some time to calm down and it's just winding them up being around others.
I honestly don't think it's that different from "normal" parenting at all. I think most people lean towards teaching/guiding anyway these days rather than harsh punishment with no explanation, that would be considered draconian. So UP just takes the same route as "traditional" parenting but instead of tacking a punishment onto things just to make sure, the idea is that you look and think, is this really teaching them anything, and could I teach them anything else in a different way. Is the part which looks like a punishment constructive in other ways, and is it really necessary? Is there a way I can teach this lesson without it being about power?
Trying to think of an example which most people would perhaps disagree with... OK so if a child is hurting another while playing, of course you remove them right away, but perhaps you decide they can't play with that child at the moment alone, so you join in, or go to play with them yourself somewhere else (or even just with another toy in the same room). Many would disagree as it's giving the child attention (and neutral/positive attention at that) even though they've misbehaved, but the UP perspective would be that actually you're teaching them to move away from someone who is annoying them and be with someone else, and/or modelling good co-operative playing skills by playing yourself, and you're more closely watching/involved so can manage any conflict before it gets to the stage of one child hurting another. And a time out (in UP view) is counterproductive because it just makes them feel hard-done-by, and doesn't really show them any positive ways of being with the other child.