Your ethernet adapter should not give you a faster connection unless you have a speed over 10mbit and the onboard isn't 10/100. And unless your computer is really old, it should be 10/100.
As far as your problem, it's going to be in your router settings. Check to make sure it's not set up to block ports that you are using or IRC in general.
Btw, Linksys isn't that good of a brand of router, imo. If you search these forums, you'll find all kinds of problems people have with various things working in mIRC. Other brands tend not to have the same problems. That's just my opinion, of course.