But yeah, you can't auto-pilot against Snake at all. You have to be conscious of his options. His mines and c4 both start combos on you really well and can control entire sections of a stage, be it Battlefield platforms that you're now not allowed to utilize or forcing aerial approaches on FD. His Cypher OOS is as fast as your Shine OOS and can be cancelled into aerials and specials after 10 frames. This lets him get free C4 sticks, nair grabs, and dair grabs, as well as start up air juggles. Besides that, his grenade covers portions of Snake's shield, meaning you're also not allowed to hit those parts of his shield without getting comboed for it and that portion changes depending on whether he's holding the grenade or if he dropped the grenade by holding shield.
His chaingrab on you lasts from around 25 until 80 percent and he has KO setups after that, and he has ways of chip damaging you up to that percent and can use up tilts to make the CG work at lower percents. No DI is best DI at low percent as it forces pivot or turnaround grabs, and DI toward center stage at higher percents because forward air and back air are both guaranteed and send you off stage. C4 Stick and Up Air are both super easy if you don't DI at higher percents, though.
Edgeguarding is incredibly easy with down air, back air, and forward air, and Cypher eats your side-b entirely and combos into back air. Falco's one boon is the fact that he can zero to death Snake, especially if that Snake isn't SDIing properly. You're forced to space meticulously and play safe against Snake and are rewarded hugely for it, essentially. Platforms aren't an escape either if the Snake is fast enough with jumping up to them or simply uses the grab walk to manipulate you into DIing a certain way. From there he can either continue to influence your DI while CGing you or shift into a down throw no-techchase (you can't tech down throw) where he can theoretically zero to death you. My point here is getting grabbed by Snake probably means you're gonna lose your stock.
I personally think Snake wins this MU and generally has an easier time going about it as well. There's just so many things that Snake stops Falco from doing. You get next to nothing out of lasering besides limiting what snake can do (but not by much, it only really affects grenades and dashdancing which aren't that good against Falco anyway) and meanwhile you can't move easily, get edgeguarded harder, and get punished way more easily than Snake. So you lose the neutral, and struggle to break even in the punish game because Snake's reversals are just as strong—maybe stronger?—than yours.