Coming in from the falcon boards :D
Falcon has very few to no approach options, but has fox beat in one area, FOX CAN'T OUTRUN DIS GAI
Falcon can beat laser spam by rushing in with a stutter usmash, a falcon kick, or raptor boost. Space properly to avoid usmash, powershield or jump > dair falcon kick, and shield to LOLOWN raptor boost.
Falcon's grabs are quite good, actually, he has a psuedo chain grab on fthrow if you don't respond fast enough. You can fair right out of the throws though, unless the CF you're fighting is Falcon Dive happy.
Speaking of which, Falcon dive is a grab type move, so it'll cut right through your attacks with grab priority. It has incredible knockback, and stage spikes if you get caught on the ledge with your pants down. CF's also have a true combo from cancelling nair after first hit to falcon dive, or failed knee ground cancelled into falcon dive. Be aware.
Falcon punch shouldn't be seen often, but exercise caution when using illusion to recover, or you may end up flying straight into the fiery fist of justice.
The knee SEEMS to have gotten nerfed, but in actuality any good falcon will know how to land it. Don't underestimate a falcon jumping at you, a bad spotdodge could lead to getting kneed. Falcon's also love to jump over their opponent and bair them, which is a fairly safe approach. Another true combo is dair to knee... it works at higher percentages, but you still need to be careful if the falcon starts autocancel dairing.
You CAN gimp falcon's Falcon dive, but considering your options to do so, its very dangerous, because of the risk of getting WTFSTAGESPIKED AT 17% and dying. Raptor boost is also a common recovery, theres a mock waveland on curved stages, and its spikes should it land in the aerial format.
Oh yeah, Falcon kick loses its lag when it flies over the edge, watch out or you might get edgeguarded off of a falcon kick off the edge.
I don't think Falcon can hold a candle up to projectile users, but fox's has no stun. Fox doesn't seem to have a stunning range advantage, so Falcon has a better time
I'd say Fox 60:40