Generally asan+msan detect more errors, and some quite significant ones. For design reasons, valgrind cannot detect most stack oob bugs. The only issue with msan is that it's a bit trickier to use, particularly if you have lots of dependencies.
Both approaches have their right to exist, but I have been preaching to people that valgrind does not find all memory safety bugs that modern tools can find, and they should be testing with sanitizers. I still experience that many people are not aware of asan+co and only know valgrind.
ASan will have false negatives on the memory accesses performed in the code which isn't compiled with ASan. It works fine, otherwise. FWIW, UBSan (-fsanitize=undefined) follows the same rule. For this reason it's common to combine them and build once with -fsanitize=address,undefined. Don't forget to pass that while linking too.
ThreadSanitizer requires that all code with atomic accesses be instrumented.
MemorySanitizer requires that all code which writes to a byte in memory be instrumented.
