Read the docs and everything you find online for that make/model of ADSL
router. What you're looking for specifically is whether it is designed
to use an "admin PC". Some of these consumer routers force you to enter
an IP address of only one machine that can log into the admin interface.
Others simply store a cookie on the first machine you log in with then
deny access from any machine without that cookie. There are probably
other variations on this theme as well.
Given what you describe, I'd be 99% sure something like this is what
you're up against. Clear the cookies cache on both machines, then try
the "problem machine" first. In addition to reading all docs you can find.