Another consideration - if you set up DHCP leases for your old rig to have a static address, you'll want to go update that by deleting the old lease and assigning that old IP to the new MAC address.
Other than that, it's a plug-and-play provided you aren't MAC filtering etc as mentioned previously.
If you haven't really done any customizing on the router's configuration, you can go lazy mode. Factory reset the puppy and set it up again (takes all of 5 minutes)