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Who wins?

Now, this is hard.
Nightwing has been trained by Batman for years and years. He was the original Robin, before he grew up (both literally and figuratively) to become Nightwing - the dark knight for Bludhaven. He is more acrobatic than Batman, and has a profound understanding of a myriad martial arts.
Batman Beyond (Terry) on the other hand does not have as much of the stringent training that Nightwing has. He cannot even claim to masterly of even one martial art, although he can kick ass with the best of them. However he has his suit. Cutting edge, that gives him capabilities beyond any utility belt, enhanced strength, boosted speed, and perfect reflexes. It also has a whole host of 'surprises.'
In a mano-a-mano fight, with no suits, Nightwing would beat the crap out of Batman Beyond. However with his suit Batman Beyond becomes a contender.
It should also be noted that NightWing was never chosen by Batman to be his successor. Even during the Bane situation it was Azrael who was chosen. Nightwing did not even know until Robin (Tim) told him that it was Azrael who was wearing the mantle of the bat.
Batman Beyond, on the other hand, was chosen. Finally Bruce considered him worthy of the mantle (although at first Bruce had apprehensions), and also he was chosen by the 'bat.' As in it was a bat that led him to the Bat Cave.
Basically Terry was chosen in the same way Bruce was decades ago. By a bat!
Hence, even though Nightwing has better skills, I'd still have to give it to Batman Beyond (who has better capability).

I thought Bruce asked Dick to take over at first, but he refused, which is why Azrael got the spot.

Nightwing never had an idea until he came to Gotham. And he was hurt that Bruce never asked him, and he thought he should have been the one. Although Tim asked him if he would have accepted and he said no,but he would have still have wanted to be asked.
For some reason Bruce never called on Nightwing to take up the mantle of the Bat.

spetnaz is quite right. Here is how the conversation went:

Nightwing: He asked someone to fill in for him?
Robin: Jean Paul Valley- formerly known as Azrael.
Nightwing: And he didn't ask me?
Robin: Would you have accepted?
Nightwing: If he needed me.
Robin: All right- but would you have wanted to accept?
Nightwing: No.

Terry Mcguiness is a pretty skilled athlete even without the suit, but he's no match for Nightwing. With the suit however, I think he could win. He's defeated some pretty formiddable foes.

Yeah, that's it in a nutshell.
Without his suit Terry is basically what Dick was while he was Robin (that is before he became Nightwing). And thus Nightwing would beat Terry (without the suit) rather easily and quite thoroughly.
However, with the suit Terry (as Batman Beyond) is basically an enhanced version of Nightwing. Their costumes are even quite similar in appearance, apart from color (they even have extendable wings). It is just that Terry's costume is completely avante garde! Almost a mini-version of the Ironman outfit! Terry with the suit is just too powerful for Nightwing.
Thus:
Batman Beyond would trounce Nightwing with suits on.
Dick Grayson would beat Terry McGuinnes without suits.

Well Night wing is more experienced, but on the other hand Terry has better technology, and also learns fast.

I think its going to be very close match.