The Wolverine vs Spider-man joke is long running but it's not very funny. There was a time where Spider-man b!tch-slaps him across the room after he sprung towards Spidey with more battle experience than Cap according to his bio. Now he's a guy who would murder Venom and Carnage while blindfolded. Well unbreakable bones and a healing factor that can regenerate from a single cell do make for a guy who's hard to beat.
The problem for Wolverine is that when you put his weapons up against Spider-man's you have a mismatch as Spidey's webs are ranged and Wolverines claws are close-combat. The one thing that has remained consistent with Spider-man is his accuracy and versatility with his webbing. He regularly hits targets at long range without even trying while web swinging and does this while thinking about other things. When concentrating he has caught small objects with the webbing while he and the object are both in midair.( his fight with kangaroo man) Kain, who had a spidersense, was webbed in the face so Wolverine will not dodge webbing without PIS. Spider-man can stay out of Wolverines strike range and pour on the webbing, and Wolverine will eventually be rendered immobile cause he's not strong enought to bust through thick webbing when one webline can support a cars weight. A webbed up Wolverine can have a webline attached to the cocoon that binds him and he can be dashed back and forth until he's unconscious. That would take a while but the end results would have Wolverine the loser. There's no reason that Wolverine could consistently negate webs that are proven to be more effective than the trapsters glue.