Fuzzy logic is good for fuzzy things. Dealing with nature, in the widest sense of the term.

Category theory is good for making extremely watertight conceptual arguments for discrete properties being preserved under transformations: it's "general abstract nonsense", yes (a famous characterization of the subject) but it really does elucidate the underlying structures in common across a lot of different mathematical settings.

If there's something wrong with the computational flavour of category theory, it's probably that it lacks fuzziness and is too much part of the linguistic/logical picture of the world. Giuseppe Longo, a category theorist, has a paper about this: http://www.di.ens.fr/users/longo/files/PhilosophyAndCognitio...