Yoshi is where you sandwich Mario enemies between eggshells. Yoshi's Cookie is where you line up cookies in a row or column. I don't remember Yoshi ever having a SNES version (there was NES and Game Boy), whereas for Yoshi's Cookie there's an NES and SNES version. I never played the NES originals, but I'd guess the SNES versions are just upgraded graphics and sound versions.
I always preferred Yoshi's Cookie (SNES) because of the title music and some other good music. Plus, whenever you finish a stage (every 10 rounds), you get to see a short animation of Mario trying to catch that darn block (or cookie or whatever) that keeps rolling away.
Thinking about it now, "Yoshi" is a pretty vague name for a game. Doesn't give you any idea what the game is, you sure wouldn't expect a puzzle game out of it, and I never thought it particularly had a lot of Yoshi for a game named after him. He's in the corner and he appears after you sandwich enemies, but that's it. Yoshi's Cookie is pretty much the same way, he just stands there moving the cursor left and right. But at least we got a 'meep'-ing Yoshi from the commercial.