Ultimately, you are correct, but Wolverine puts up a serious fight. He would dog Spiderman all over town, sniffing him out, never giving him much time to rest.
Spider strength would not overpower Wolverine. He has faced enemies far stronger and had no trouble.
There would be no rest as Peter Parker. Wolverine doesn't need the costume to tell that his enemy is near. But Peter's science background could help Spiderman concoct a sleep potion or alter his web shooters (which Wolverine would cut through as they are).
But Spiderman's blather - I mean banter - would eventually be one of his greatest weapons against the stoic Wolverine. This would confound and enrage Wolverine to the point where he might be lured into a trap.
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"Jelly-belly gigglin, dancin and a-wigglin, honey that's the way I am!" Janice the Muppet