This was a somewhat hotly debated question in the 1980s. The debate was more-or-less ended with papers like Cheeseman's In Defense of Probability.
The short answer is that Fuzzy Logic does not just assign a continuous value to sentences, what it does is assign degrees of membership in different fuzzy sets. These degrees of membership range between 0 and 1.
In contrast, probability says: Among the set of values this variable could take on, what fraction of them are in a certain set? This fraction also ranges between 0 and 1.
This might seem like a semantic distinction, but it has deep implications if you try to use these systems to reason about uncertainty.
For example, consider the question "Will it rain tomorrow?". A probabilistic approach would try to determine the fraction of days like tomorrow that have had rain. A fuzzy logic approach will try to determine whether tomorrow is like a rainy day. The distinction becomes obvious if we then ask whether it will not rain tomorrow. The probabilistic approach will try to find the fraction of days like tomorrow where it did not rain. The chance of it raining or not raining will sum to 1. The fuzzy logic approach will try to determine whether tomorrow is like a non-rainy day. Note that tomorrow may be both like a stereotypical rainy day and not like one. There is no firm requirement that these sets be disjoint. This reflects Cheeseman's critique of Fuzzy Sets: they implicitly reject the additive axiom of probability theory, which (as denoted by the name axiom) is something that is fairly unreasonable to reject.
While there are various approaches to make memberships in fuzzy sets more probabilistic in nature, that's the root distinction.
I think the dominant modern view is that fuzzy sets are a great tool when you need to reason about membership in a fuzzy concept. Whether rice is "done" or not isn't a question of something being true or false. It's a degree of membership in the set "done". On the other hand, whether the object in front of a self-driving car is a person or not is either true or false. It is not necessarily a good idea to reason about this in terms of whether the object is partially in the person set or plastic bag set.
