That's really odd, but one thing is certain: The problem revolves around something that is active when online only. Because of that, I'd suspect driver or IRQ sharing issues between your modem (or network card) and your sound or video card(s).
I've thought about it for a while, and I do not see how changing a monitor could possibly cause this. Monitor .inf files are simply a list of resolutions and refresh rates supported by that model of monitor. (There's also a name and possibly some color information in there as well.) Certainly nothing that would depend on you being online or not. To rule out the possibility though, I suppose you could reconnect the old monitor and see whether or not you still have the problem.
One old issue that could be the problem is the connection icon itself in your taskbar. Some video cards (especially Voodoo 3s, and some ATI cards) would stutter badly as those little lights flashed on and off. Removing the icon would fix the problem.
