A Faster computer does not equal faster Internet
Just as Rick mentioned that people often expect the computer to run faster once they have faster Internet, the opposite is also expected.
Often people on dial-up or slower DSL ask me once they have a new computer whether their email will go faster or their websites will load quickly. Typically the difference between a new computer and a Windows 98 machine with sufficient RAM will be barely noticeable.
It may take longer for programs like Outlook Express or Internet Explorer longer to start up on an older machine, once the program has loaded the actual reaction time for the Internet is pretty much unchanged.