|A little unclear here as to what is where...|
Are you switching a single wifi adapter from one computer to another - as in you do not have a wifi card/adapter in each PC? Or do you mean you have two identical wifi cards - one in each PC (desktop or tower)?
What make and model is/are the card/s.
Does the working PC see the SSID for your wifi signal from the router: does the "other" PC see it or not?
Have you set the option to "hide" the SSID in the router's wifi setup/management; and if so - do you actually know what it is (and the password for it)?
On the router - have you enabled any form of access control - apart from an encrypted access key/password?
Have you set up MAC filtering on the router - which allows any one of these options: allow anyone to connect (to wifi); allow only those in the list to connect: disallow those in the list to connect...
Edited to correct syntax error...
message edited by trvlr