I can't imagine how a fast connection would help, seeing as how, when you're taking the shot, everything should be running solely off what is already downloaded and running in the memory of your computer.
There are any number of reasons why one computer would be better than another at WGT, though. To name a few...
(1) how much junk is running in the background, especially auto-updaters and anti-virus software
(2) mouse type
(2) quality of mouse driver
(3) processor type and whether WGT works better with, say, a two-core high frequency processor, or a four-core processor with a lower frequency (the answer to that is almost assuredly the former)
(4) operating system; certain operating systems are more responsive than others, meaning that they will check more often for input from a device such as a mouse, making the latency not only smaller but also more consistent