Depends very heavily on the task and the string implementation involved, but my prior is that you should always bet on regex for anything more complex than "ForEach chr in str: doSomething(chr)".
Modern regex implementations are their own language interpreters/compilers, they implement their own string data strutures, they compile your regex into specialized bytecode (different from the hosting language), they have tons of special cases checks for fast paths (e.g. capturing groups make regexes much much more complicated than they need to be, a regex without capturing has the option to run signficantly faster).
A random search just found out this 2016 SO question[1] where a compiled regex was faster than string.contains (!). Regex is just insanely optimized. It reminds me of C++, ugly and badly/un designed, but it can go fast.