I'll give X4 to Captain Kirk VS Han Solo, X3 to Finn & Jake VS Mordecai & Rigby, and X3 to Harlem Globetrotters VS the Monstars.
As for this match, it's tricky. The first Staypuft was a manifestation by Gozer, the evil God, but they have reappeared separately, it seems, mainly in video game boss fights and the Real Ghostbusters cartoon. In the SEGA Genesis game, he's basically elastic, can shoot fireballs and eye lasers, and manifests to serve a different Goddess out of another ghost. In the cartoon, he manifests out of a boy's dream, and seems to be a wholly separate entity from any Gods, mostly. He is at first an antagonist, but sometimes a partner to the reluctant Ghostbusters, helping them fight off other giant ghost-possessed monsters such as Murray the Mantis. There are a lot of incarnations, it seems, I dunno if I'll list them all out. I know the more recent Ghostbusters game had him also able to make little Staypuft clones, but he was also in a weakened state compared to the first movie. In some incarnations, mainly the IDW and Real Ghostbusters comics, it seems like you have to destroy him on a molecular level, especially given he is some kind of phantasm rather than a regular being. The Ghostbusters have specialized equipment for that, hence why they were able to blow him up, though it became harder when he returned in some instances. In general, he is also a giant stomping being who can throw around marshmallow goop. A lot of this info was aided by the Ghostbusters Wiki.
The Jolly Green Giant...he seems more of an intelligent being, or at least able to converse like a human usually can more consistently. His origin doesn't really have much of an explanation beyond "he wanders around crop fields and enjoys vegetables". What I can tell you is that the guy is ridiculously strong. You may be thinking "well, of course, he's a giant", but that's not even the half of it. He
scooped stars from the sky,
forced the sun down to make it night time,
broke off a large chunk of the moon and turned it into a small river of liquid cheese,
held the moon again to create light for his veggies to grow,
can casually hold lightning, and he can basically use magic,
turning the moon into a flashlight and, via his can,
teleported this poor kid into his alternate cartoon reality. Mascots really enjoy doing things like that for some reason. All of this to say, I think in terms of attack potency and hax, the Jolly Green Giant has it covered.
Can the Jolly Green Giant use his magic to molecularly destroy Staypuft, if we count all of Staypuft's incarnations together? I am unsure. Staypuft definitely isn't doing much to harm the Green Giant, from what I know. If it were Gozer themselves, then maybe, but the Staypuft manifestation, maybe not. One method that could potentially be used: the Green Giant could potentially just use Battlefield Removal on him, or BFR, which is somewhat similar to how the Ghostbusters bust ghosts, and trap him inside of a veggie can. However, given that in certain incarnations, we see the Ghostbusters able to just release him again, this might not fully count. The question remains if the Green Giant could transmutate the Staypuft Marshmallow Man into something more harmless. I feel like if, say, he got turned into a corn cob, maybe that would count as a win, but even then, via his residue being left over in some Staypuft Marshmallows, that has resulted in another ectoplasmic entity or some such in the Real Ghostbusters cartoon, known as Monster Munch (a living sandwich, not a British snack food). So even if this happens, the Green Giant may be inadvertently creating another enemy for himself.
So, my thinking is this: Staypuft is not as strong as the Jolly Green Giant, and the Jolly Green Giant has magic that may be able to destroy him in a similar manner to how the Ghostbusters did, but I don't know for sure. However, it is the only way I can think of for one to kill the other, so I will say that the Jolly Green Giant wins this for the time being.
