It's time for a good old-fashioned Marvel-vs-DC showdown, folks!
Both are bouncy, martial-artist types, more used to shadows and skullduggery than outright bad-assitude. Spider-Man would win in feats of strength, because come on, what's Dick Grayson got, the proportional strength of a robin? Please.
Nightwing has the edge in actual martial arts training, though. You don't study under the GD Batman for that many years without getting pretty darn good.
Both can invent clever gadgets to help in the fight, and both have plenty of real-world experience mixing it up hand-to-hand.
Spidey has the enormous advantage of his Spider-Sense, though if it happens in a karaoke bar, you'd have to think Nightwing's Robin's Song would come in handy. If only that existed, right?
At the end of the day, though, I have to go with costume design, and on that score, Nightwing wins. I always dug the cool black and dark blue motif. Had he kept the doofy ponytail, I'd dock points, but he seems to have come to his follicle senses in time and gone clean.
But that's just me, who would you choose?