In the beginning (well, not quite the beginning) there were 2 internal architectures - Windows (as in 3.1 and it's successors) developed solely by MS and the NT architecture developed by MS in a joint collaboration.
Windows 2000 was the last solely NT architecture but windows XP, although it brings the two architectures together to some extent, is really an NT architecture successor.
The answer to your question is very much - 'it depends'. XP was designed to be able to use both windows and WinNt drivers - which it did to an extent. As result, any driver written for the NT/2000 architecture may run well on XP and the manufacturer may not have updated it significantly. In these cases the XP driver may well run on 2000. If the driver was written specifically for XP then it is unlikely that it will run on 2000.
I haven't asked why you want to avoid buying XP - is it just cost. It is only my personal opinion but I think XP was the the best quality OS to come out of MS so far. Maybe Vista will eventually become that but I think it has a way to go before it overcomes all it's bloatware.
One thing you may be interested is that, from using 2000, the internal directory structure and architecture of XP will mostly be familiar to you. With Vista it is very different. There seems to be a virtual directory structure overlaying the real one and I found this quite hard to get to grips with.
In terms of your PC running slow, I would say that you should try to have an OS of the same generation as the PC. This way the the power requirements are likely to be matched by the equipment that was being sold at the time.
If you want to go for XP then you should be able to get a very good deal on a genuine version if you can find anyone still stocking it.