Your computer's processing power does affect connection speeds if you are transferring large files over a LAN, like to transfer a couple of gigabytes from 2 dual core machines over a LAN would transfer faster than between 2 fairly older machines. I mean it wouldn't take that much longer but that's about the only time you would notice, otherwise as far as internet connection goes, if you were to download a 10mb file off a website, any computer, especially anything fairly new should be able to download the same file as fast as the next one, providing your using the same internet connection.