Methodologically, one story they did clash, but Zeus broke it up with a thunderbolt before it was resolved. Apollo was one of the most popular and diverse Greek gods up there with Hermes and Zeus. At the same time Herc was the Greek superman and pretty much a god in mortal form. For the mythological versions I would give it to Herc. His whole shtick was to be invincible in direct battle. He defeated Ares, Hades, Thanatos, and I think even held his own against Poseidon for a while in one story. Heck, in some books (but I've yet to find the actual mythological source) it states he wrestled Zeus to a draw. However, I imagine the Greeks would see it as more epic than Herc vs Ares. Given Apollo's popularity and how high up he was suppose to be that would explain why there were never any stories of a true winner between the two.
Comic versions, depends. Herc is obviously too much for Apollo in a straight physical fighter. He is a stronger and better fighter. Apollo is probable a better archer. If Apollo were to use his solar powers effectively I could see him winning his fair share even if he did mix it up to a degree physically.
Yeah I have read the big about wrestling Zeus. But that was after his mortal death. On the other hand when Zeus challenged every single God against him to a tug of war, everyone was silent. By this time Heracles should already reside in Olympus, I think.
So, was it a case of skill or actual power? It`s never mentioned. But Heracles wrestled against Thanatos as well and had him flee, so I am inclined to think that wereas his strength was vast, his skill as was vast on itself.