This review mostly compares Log::Handler with Log4perl, which is a mature and one of the most popular logging frameworks.
I think Log::Handler's interface is much simpler, nicer, more Perlish than Log4perl. It's a bit similar to Log::Any::App, which I created just because I hate Log4perl configuration.
There is a unique concept of maxlevel not normally found in other frameworks, though it can be emulated in other frameworks using filters.
At a quick glance, the speed is around twice that of Log::Log4perl, so I'll say it's on the low-end side (there are other much faster logging modules, but anyway speed is not an issue to most people).
It currently lacks sublogger (hierarchical categorization and adjustable/automatic appending of subcategory to its parent), so it cannot be used to replace Log4perl in most cases as that's one of the main feature of Log4perl. Which is a pity because I would otherwise switch.