mod_perl2 is really difficult to beat if you need to dig into the guts of Apache. DBIx::Class is a huge win if you need/want an ORM but you've been given a really weird pre-existing DB schema. POE has been around forever, and makes really stable long running daemons that don't have to be babied. CPAN has stupid amount of tools for automating sysadmin tasks, all of which you'll probably end up using once, putting in a cronjob somewhere and forgetting about because it will just work. Any time you need to deal with something old, or weird, or both there's probably a CPAN module for that, you can find it via search, and it probably has tests and docs. Template Toolkit for creating uglyass text is both nice to use and full featured (if you don't want to go down the XSLT hole). I've also found that Perl handles I18N, unicode and uglier things (like shift-jis) fairly well. Almost every payment processor has a module, and they all have a standard(ish) API, so they're mostly interchangeable. Likewise crypto stuff is fairly standardized and pretty complete. In general, if you have to tie something to something else that's weird, Perl is a good bet (because someone has probably done it, put it on CPAN, and documented it).
That's a few of the problems I think it stands out as a great tool for.
mod_perl2 is a brilliant tool for writing apache modules.
Catalyst is a far better web framework. So's HTTP::Engine.
And with the PSGI standard and Plack reference implementation, the lighter weight stuff is starting to standardise too - my new toy, Web::Simple, leverages Plack to provide Catalyst::Engine-like functionality but in a very lightweight manner and very shareable between different frameworks with different philosophies.
I remember showing DBIx::Class to Simon Willison at a LugRadio Live - he looked at some of the complex queries chained resultsets and easily build and his jaw dropped. We were going to try and get round to me helping him fix Django's ORM to have more of the capabilities of DBIx::Class in return for him helping me build a django-admin-interface-like piece of code for Catalyst but we never quite got to it, sadly. Happily, our respective communities seem to have already done similar things without us :)
I Absolutely agree with regards to web frameworks. But there's a time and a place for apache modules, and mod_perl is definitely the tool for that job.
That's a few of the problems I think it stands out as a great tool for.