They are primarily of value for highly structured text parsing. If you used named groups (and option in most mature regex systems), you have a phenomenally powerful and crisp way to handle the strings.
Here's an example. Consider that netstat in its various iterations on different linux OSes, and versions of netstat can return different results. Sometimes there is an extra column, sometimes there is a shift if the date/time format. Regexes give you a powerful way to handle that with a single expression. Couple that with named groups, and you can retrieve the data without hacks like:
1) split on spaces
2) ok, the netstat version is X so add I need to add 1 to all array references past column 5.
3) ok, the netstat version is Y so I need to make sure that I use multiple array references for the date info.
YUCK. Simple to fix in a Regex :-)