I know it has its own symbol in Super Smash Bros and is classified as it's own thing in said game, but is it really seperate from the main Mario series?
Because Bowser's been the main villain in all the Yoshi games in existance (along with Kamek), standard Mario enemies regularly appear, various hallmarks of the Mario platformers often appear (warp pipes and what not) and the first game was even called Super Mario World 2 in some regions.
Is there enough of a difference in the content of the Yoshi games to count them as distinct from the Mario franchise in general?
Oh dear lord no. Just because they have a diffirent playable character with slight changes to playstyle doesn't mean it's a diffirent series entirely. What next? Calling MGS2 a diffirent game because you play as Raiden?
Well, the Wario series and the Donkey Kong series all have different villains besides Bowser. Yoshi games generally play different than Mario games, though. And... anyone remember this game?
I would call the Yoshi series seperate. I mean Mario was not in Yoshi's Story. He was cameoed, but never seen. Plus Wario and Donkey Kong all live in Mario's world. (At least I think DK does. He kind of just shows up out of no where now a days.) Either way, if they can be there own series I don't see why not to call Yoshi a different series than the Mario series.