Welcome, Guru~
Your question's an interesting one that sometimes separates people along rather firm lines. I'm definitely no expert (just 4 years of self-recording experience), but I have friends and colleagues in both camps, and they could argue for hours about the trade-offs of each approach. The answer's a personal one, reflecting your own needs, outlook, and background--only you can figure out what's the right call for you.
Some personal questions to consider:
1) Do you need portability?
2) Do you like the idea of having a self-contained unit with one manual and a forum of other users to guide you?
3) Do you mind using a computer so much? (Some folks use computers all day long at a day gig but then like to get away from a traditional computer setup in their studios while other folks just struggle with basic computers, in general. There are plenty of people, though, who zip around computers almost without thinking.)
4) Do you need future expandability, or our your needs modest? (Some folks actually like having some limits so that they can focus more on the music itself. Other folks like the technical side of things and can't wait to tweak and expand their systems.)
5) How's your money scene? If you go the computer route, most people seem to agree that a dedicated system is a wise choice, but that could mean more money than a standalone.
6) Last, and a big question, how professional do your recordings need to be? A dedicated computer approach will provide more options, especially over time, for improving your sound. But do you really need that level? A good stand-alone recorder can produce excellent recordings in the right hands, but such recorders do have a limited set of features. In the end, though, your skills will likely be much more important than which digital system you buy.
Just my own amateur take on things. As Hornet wrote, it's all about enjoying music, and either approach could work beautifully for you.
Good luck with your decision, man~
J.