What kind of bugs do you find this way, besides missing sanitization?
Basically the whole world of bugs introduced by someone being a too smart C/C++ coder. You can battletest parsers quite nicely with fuzzers, because parsers often have multiple states that assume naive input data structures.
For example you can make it generate queries and data for a database and generate a list of operations and timings for the operations.
Then you can mix assertions into the test so you make sure everything is going as expected.
This is very useful because there can be many combinations of inputs and timings etc. and it tests basically everything for you without you needing to write a million unit tests
With fuzzing you can randomly generate bad input that passes all of your test cases that were written using by whatever method you have already been using but still causes the application to crash or behave badly. This may mean that there are more tests that you could write that would catch the issue related to the fuzz case, or the fuzz case itself could be used as a test.
Using probability you can get to 90 or 99% or 99.999% or whatever confidence level you need that the software is unaffected by bugs based on the input size / number of fuzz test cases. In many non-critical situations the goal may not be 100% but 'statistically very unlikely with a known probability and error'