If someone passes you a text file that is verified to be valid UTF-8 and contains, say, access permissions, then you better not stop parsing it at the first '\0' character.
None of this is a huge problem, but it's something to be aware of. C string handling is incompatible with UTF-8.
That's separate from string handling.
UTF-8 was originally designed to be compatible with NUL terminated strings and keep NULs out of well formed text.
In fact it was the first point in the 'Criteria for the Transformation Format', mentioned in the initial proposal for utf8.
The UTF-8 spec doesn't make that distinction as far as I know. There's a simple fact: A valid UTF-8 byte sequence can contain nul characters. So you can't naively use C string handling functions on it. And as someone else has correctly pointed out, the same is true for ASCII.
I'm just pointing out a potential pitfall and a source of security issues. Some might assume that after validating UTF-8 text input, you could just dump it in a C string and process it using C's string functions. But that's not the case.