I somewhere read that during the recording sessions for Keep The Faith Jon played stones videos and concerts all over the place. And when you listen to songs like Sleep, Blame it on the love or the bassline of KTF you can really hear that influence. But in overall I wouldn't say that there is much Rolling Stones in Bon Jovi songs. Sure with Richie there's a bluesy kind of impact since he has some of the same guitar rolemodels as Keith Richards has but that's not enough to make the songs comparable to the Stones sound.
They're both rock & roll bands with roots in the bar and club scene. That's probably as far as it gets.

Someone mentioned Bitter Wine as being Stones influenced. When I listen to the electric version of that I would rather say it's a second With a little help from my friends kind of song (especially the intro). They probably ripped it down to an acoustic ballad afterwards.