Bayesian probability addresses uncertainty about whether an event will/did occur, but treats the actual occurrence of the event as a boolean value of {0, 1}.

Fuzzy logic recognizes degrees of occurrence, represented by any number between 0 and 1. Its proponents contend that this allows richer abstraction of the ways that humans actually perceive and reason about the world. In their view, probability is just a special case of fuzzy logic, using artificially restricted truth values.

If you're interested, check out Kosko's "Fuzziness vs. probability." International Journal of General System 17.2-3 (1990): 211-240 http://sipi.usc.edu/~kosko/Fuzziness_Vs_Probability.pdf