Why use bzero over memset?

I don't see any reason to prefer bzero over memset.

memset is a standard C function while bzero has never been a C standard function. The rationale is probably because you can achieve exactly the same functionality using memset function.

Now regarding efficiency, compilers like gcc use builtin implementations for memset which switch to a particular implementation when a constant 0 is detected. Same for glibc when builtins are disabled.

I'm guessing you used (or your teacher was influenced by) UNIX Network Programming by W. Richard Stevens. He uses bzero frequently instead of memset, even in the most up-to-date edition. The book is so popular, I think it's become an idiom in network programming which is why you still see it used.

I would stick with memset simply because bzero is deprecated and reduces portability. I doubt you would see any real gains from using one over the other.