In Melee she had Fair and Bair thus she had a much better air game which helped her A LOT. Approaching was also a less big deal than it was in Brawl, making one of her weaknesses smaller, not to mention she had a better air game that also helped her. She was pretty bad at comboing, but she had high kill power and she was one of the hardest ones to combo. She was still very vulnerable to camping due to her bad mobility, but at least she had wavedashing, even though her wavedash was not so good.
Her smashes are a bit worse. Her Usmash is no way as good as Brawl, Her Fsmash seems to be a littler weaker in Brawl (nor sure), but it was pretty easy to DI out of in Melee and her Dsmash was weaker.
Her Dtilt is worse in Melee as it is slower and doesn't 'stun' as much as in Brawl. Her Ftilt was better in Melee, because she could at least combo with it and kill at higher percentages. Utilt is a bit weaker than in Brawl, but you could start some combos. Her Dash Attack is pretty much the same I think, not sure about this. Her Jab had more range in Melee and it may also be faster, I don't know about that. Jab just feels better in Melee ;p.
Her air game is a lot better in Melee. Fair and Bair are lovely! You could survive on those two alone. Uair is weaker, but has more range. I think Dair has a bigger sourspot but smaller sweetspot, not sure about this. Nair is somewhat worse, as it has less range and easy to DI out of.
Nayru's Love's last hitbox doesn't connect as often as in Brawl (which is bad, because the opponents won't get hit away and can punish Zelda), but Nayru's Love seems to have more range. Dins Fire is even slower and has a smaller hitbox. It's even worse than in Brawl. Farore's Wind covers more distance in Melee but it doesn't have that last hitbox as in Brawl.
I'm not sure which one is better, especially because I know very little about Melee and Melee Zelda, but this is what I've observed, and I think that Melee Zelda is a bit better.
Just adding to this because I played Zelda in Melee.
Zelda had the worst wavedash in the game (tied for worst distance and 2nd slowest). It didn't help her mobility much (you're better off dashdancing or walking in most situations) but was useful occasionally for spacing. Wavelanding is a better spacing tool. In any case, Melee's spacing tools helped her land kicks more often.
Her grab was about as bad, but her throws were a lot better. Uthrow was a chainthrow on fast fallers. She was one of the characters that didn't benefit AT ALL from JC grab, because her dash grab was the same speed but with much more range. Speaking of throws, her throws punished anyone with poor DI.
Dsmash, while it seemed to be less "powerful" also benefited her more in Melee because of crouch cancelling. It came out on frame 3, making it faster than her jab. It also sent opponents at a 45 degree angle down. Against many characters in specific situations, crouch cancel dsmash was useful. If someone didn't sweetspot the edge or tech correctly, a crouch cancelled dsmash could gimp at ridiculously low %'s.
Dair had no "lightning kick" and was probably the worst dair in the game. You can up b stall on the ledge.
And, her down b was infinitely better. Having trouble with your opponent? Instant top tier!
Characters get attention not so much by placement but typically by how many fanboys the character tends to attract. Link, Ike, and Falcon are pretty much the triforce of Fanboyism.
What about SONIC?? He's a huge fanboy character.