Bayesian statistics has to follow Bayes rule. There is a fairly specific framework behind the numbers in Bayes (i.e. probability). Fuzzy logic is looser.

You could say Bayesian statistics is a subset of Fuzzy logic.

Given how informal people have to be in Bayesian statistics to come up with reasonable priors (e.g. uniform), and how well it works by just guessing reasonable values, it could be argued that the power of Bayes is not in the inference but from the slack in the system it permits. Fuzzy logic is pure slack.

I think modern neural networks with activations like leaky relu look more at home in a fuzzy logic textbook than in a statistics text book.