A bit, but while they're related I think they're best considered separately. While both are figurative and express that the speaker would be prepared to bet money (or something else) on the possibility addressed, they do so in slightly different ways.
"I'll bet it works", is claiming (though only figuratively) that they'll actually make that bet. Contracted to "I bet it works", it sounds almost looking for takers and offering odds (not a change in meaning, only in nuance).
"I'm willing to bet that it works" is a similar claim of confidence, made in a less informal manner (though still not very formal speech).
To me that reads not so much of less confidence, but as less emphatic. And the difference is more between the sort of thing that makes more sense as an exclamation or in direct addressing a particular person, over the sort that works both there and in a description or narration. However:
"I'm willing to bet all I own that it works", would be stronger than any so far, as would figuratively offering absurd odds: "I'll give you a million to one that it works", "I'll give you a guinea on a farthing that it works".