How is high speed differs from dial up internet?

Speed for one.
Dial up has a maximum speed of 56 Kilobits, where as High speed or Broadband normally starts around 128 Kilobits and maxes out at about 24 megabits (current ADSL2+ speed). Of course there are much higher speeds out there but they tend to be used by large companies and ISP's

Another notable difference is that dial-up uses your current phone line whereas high-speed is separate from from phone service. With high-speed you can be on the phone and on the internet at the same time.

And a third difference is that broadband is a constant connection.

There are some dialup schemes that bill themselves as high speed. They use caching schemes to speed up your connection. But mostly high speed refers to broadband