Just going to expand a little on what Sean says:
Falco/Peach swings back and forth between which character has an advantage against the other, and its 100% based on how much experience the PLAYERS have versus the other character.
A scrubby Falco will get **** on by Dsmash.
A scubby Peach will have absolutely no idea how to get around Falco's "stuff"
As each player becomes more aware of what the other character can do, and how to deal with each others tools, it becomes more in-depth.
I feel that Peach has an advantage over Falco on Brinstar, Final Destination, and Battlefield. Each of the stages compliment Peach's strengths, whilst providing various stage-related hazards for Falco to deal with. Brinstar, Peach can use the stage to her advantage more easily, whilst its extremely easy, even for good Falcos to have to essentially fight a 2v1 vs both Peach, and the Lava/Middle.
On Battlefield, the stage doesn't give any distinct advantages or disadvantages to either character. everything that helps one character, helps the other equally. The deciding factor is that Peach has an easier time edgeguarding Falco due to his limited options. Peach SHOULDN'T get gimped by Battlefield, but it DOES occasionally happen. I don't think it happens at high-level play.
On FD Falco loses the protection of platforms and becomes Peach's *****. It's extremely hard to limit Peach with lasers due to her float sweetspot, and Peach is free to do anything she wishes with Falco. With good reaction/prediction, Peach can literally grab a Falco 0 to death on this stage. To elaborate; Uthrow Dsmash does about 30%. This leads into a tech chase opportunity, which then leads into a deathgrab, assuming Peach landed the grab. If there is a reset to neutral, Peach just got a free 30%, and any grab following this should be the death of Falco. Even if the grab itself doesn't KILL Falco, he's now off the stage and recovering with high %.
In Falcos favor we have Rainbow Cruise, Kongo Jungle 64, Dreamland 64, Yoshi's Story, and Fountain of Dreams.
I won't go in-depth into RC and KJ64, I personally believe those to be, by-far, the most falco-friendly stages currently legal, and just absolutely unfair for Peach. ESPECIALLY RC. For lack of better words, that stage is just GAY.
Examining the Neutrals, we have whats considered to be the "best" stage for jiggs and peach; Dreamland.
This is a common misconception. While they might benefit from living an extra 20~30% longer, they have to deal with platforms that are just totally in Falco's favor. The height and width of the platform makes platform tech chasing harder for Peach, whilst also limiting most of her options against a defensive Falco. Offensive-wise, the platforms are just the perfect height to also compliment Falco's offensive options.
Yoshis and FoD are for more offensive-minded Falcos. Peach's CG is mostly nullified, whilst Falcos offensive options are at their utmost max on neutrals. Short blastlines off the side on both sides, and the smallest ceiling on YS. Falco's recovery is also enhanced due to the bottom of the stage just being a gigantic wall, making wall techning easy, and adding more walljump mixups.
Pokemon Stadium is a stage I believe to be 50/50. Peach keeps her CG, and has easy platform tech chase setups, whilst the stage transformations help defensive-minded players. Falco's offensive options are about the same as FD, but has the benefit of platforms for easier laser height mixups, and means of continuing combos.
Just examining the stages alone, I'd say Falco has the advantage. He has more stages that work to his favor. The difference is that Peach has two stages that are just absolutely horrible for Falco. Falco really only has one. Arguably 3 if Falco wants to play the clock, but I don't think anybody, except maybe Th0rn is going to attempt that.
As for gameplay itself; A basic summary of the match is both characters throwing out stuff just outside of the other players attack range, trying to punish bad spacing. Easily one of the most spacing-orientated matchups in Melee.
. And I want a paper chocobo!
