In terms of Frodo and Bilbo's difference in resisting the Ring, remember that Bilbo spent his possession of the Ring getting fat in Bag End. Frodo endured starvation and exhaustion, and was actually in Mordor. The closer you are to Mordor - and indeed the more powerful Sauron is - the greater power the Ring has.

Still, Frodo definitely failed his quest. Not out of a lack of trying. He was in over his head and he knew that from the start.