While I disagree with Flimzy's introduction ("I've not used Moose, but I have used this thing that uses Moose"), I agree with his premise.
Use what you feel you can produce the best results with. If the (or a) goal is to learn how to effectively use Moose then use Moose. If the goal is to produce good code and learning Moose would be a distraction to that, then don't use Moose.
Your question however is open-ended (as others have pointed out). There is no answer that will be universally accepted as true, otherwise Moose's adoption rate would be much higher and I wouldn't be answering this. I can really only explain why I choose to use Moose every time I start a new project.
As Sid quotes from the Moose documentation. Moose's core goal is to be a cleaner, standardized way of doing what Object Oriented Perl programmers have been doing since Perl 5.0 was released. It provides shortcuts to make doing the right thing simpler than doing the wrong thing. Something that is, in my opinion, lacking in standard Perl. It provides new tools to make abstracting your problem into smaller more easily solved problems simpler, and it provides a robust introspection and meta-programming API that tries to normalize the beastiary that is hacking Perl internals from Perl space (ie what I used to refer to as Symbol Table Witchery).
I've found that my natural sense of how much code is "too much" has been reduced by 66% since I started using Moose[^1]. I've found that I more easily follow good design principles such as encapsulation and information hiding, since Moose provides tools to make it easier than not. Because Moose automatically generates much of the boiler-plate that I normally would have to write out (such as accessor methods, delegate methods, and other such things) I find that it's easier to quickly get up to speed on what I was doing six months ago. I also find myself writing far less tricky code just to save myself a few keystrokes than I would have a few years ago.
It is possible to write clean, robust, elegant Object Oriented Perl that doesn't use Moose[^2]. In my experience it requires more effort and self control. I've found that in those occasions where the project demands I can't use Moose, my regular Object Oriented code has benefitted from the habits I have picked up from Moose. I think about it the same way I would think about writing it with Moose and then type three times as much code as I write down what I expect Moose would generate for me[^3].
So I use Moose because I find it makes me better programmer, and because of it I write better programs. If you don't find this to be true for you too, then Moose isn't the right answer.
[^1]: I used to start to think about my design when I reached ~300 lines of code in a module. Now I start getting uneasy at ~100 lines.
[^2]: Miyagawa's code in Twiggy is an excellent example that I was reading just today.
[^3]: This isn't universally true. There are several stories going around about people who write less maintainable, horrific code by going overboard with the tools that Moose provides. Bad programmers can write bad code anywhere.
I've not used Moose, but I have used this thing that uses Moose
? I think you have a misunderstanding of Catalyst. Catalyst does not use Moose; it is an alternative MVC. – Isochor