Quality as such wont suffer just because your computer is a bit old, but your working conditions might to some extent. The main gripe I had with my PIII was that it wasn't powerful enough to run all the DX/VST effects I wanted in real time (e.g. Waves RVerb ate 50% of the CPU overhead just like that). The workaround is simply a matter of bouncing a 100% wet track of the effect you want, and mix it in with the dry track. While this is simple enough to do, it does take some time and in the long run working like that might annoy you a bit.
With GigaStudio, low processing power will mainly hamper your polyphony. GigaStudio will work with your computer, no doubt, but it's not like you'll get 200 voices or anything. But I suppose you could do the bounce-to-track workaround with GS as well.
I guess there's no doubt that faster is better when it comes to computers. But if you're short on cash, or at least able to find better ways to spend your money than a new computer, you could try using your PIII for a while and see how it works for you. If it doesn't, a used P4 most likely will, and shouldn't set you back a whole lot.