There are two main ways to make this work:
1. Do the full system restore as you describe, then boot into VGA mode to install the correct video driver and use the Devices applet to correct the disk drivers. The NIC driver is the simplest to correct, as you can use the Network applet in "normal" boot mode without necessarily booting to VGA mode.
If the hardware is so completely different that this is not even an option (i.e., Windows won't even BOOT after the full system restore,) you are left with a scenario that will take longer to complete, but in the long run might leave you with a cleaner system overall. I actually prefer this second method for that very reason.
2. Install a clean copy of the OS onto the new (different) hardware, install relevant applications to get your registry entries correct, then do a selective restore of the DATA ONLY onto the new server - do not restore the %systemdir% directory or the system registry.
This was first published in June 2002