BRL,
I assume you mean to replace the HDD in the new PC with the HDD from the old PC.
The physical transfer is relatively straightforward, although you will probably find that the disk is installed in a "caddy" that you would need to extract it from to swop over. Possibly tricky and fiddly.
The next problem is that, unless the computers are identical, the chipset, video, network, sound drivers etc. are all going to be for the wrong PC. This really is a big problem, and is effectively almost insurmountable.
You can do a Windows Repair from the original install media (install, repair), which might work, but it is (in my experience) more hassle and less reliable than simply re-formatting and doing a clean install. Obviously you have to have all your data and settings backed up, and access to all the installation media for all the applications you want to install.
The next problem you might run into is a lack of drivers for XP on the PC that currently has Vista. It is a low probability, but possible.
The final problem as hellsbrink points out is to do with Windows validation. The original windows XP installation would see so many changes to the system that it would require revalidation.
The fact that the OS installed on the new PC is Vista has no bearing - as soon as you remove the disk there is NO OS installed!!
SD