Go to your router's web based menus and make sure every setting is correct. Also remind yourself of settings on it that you will need for Yast. For example, my playstation is x.x.x.102 and I would need to know that in the setup of the internet connections on the ps/2. Then, I would go through the card's setup in Yast.
Your router should be a DHCP server for your computer.
I understand your setup to be:
--- ---computer that doesn't work
|----telephone line from wall----dsl modem----router---/ ---------
---computer that does work
The computer that does work could have a static ip set in the DHCP server of the router... and because the new computer doesn't and there aren't any numbers to assign free it just doesn't work.
It is also possible that you didn't set your username and password right in the installation and your ISP won't let you on. That is correctable in Yast. This is less likely because you have problems internally too.
Or I could be full of it.