Someone used Nom for Advent of Code last year and found "The regex approach benchmarked at about 1ms while the parser approach benchmarked at 145 nanoseconds." [2] Maybe I'll try benchmarking Nom vs. regex for a follow-up post.
I find regexes are easier than parser combinators for simple tasks. If the problem is small, parser combinators are overkill. But if the problem gets complicated, I think parser combinators are more readable. You can break a parser combinator into small, well-documented, well-tested parts more easily than a big regex. But I still use regex more in my work, because most complex parsing I just throw into serde.
[1] https://github.com/Geal/nom
[2] https://www.christopherbiscardi.com/advent-of-code-2020-in-r...