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According to Wikipedia, the Amish (aka Amish Mennonites) are a group that broke off from the Mennonites in Switzerland in 1693, led by Jakob Ammann, from which the name comes.1
While Amish reject many modern technological advancements, Mennonites have a broader range of acceptance of such things. Some Mennonites are quite similar to the Amish, while others accept all modern technology. I had a friend in college who was Mennonite, and he drove a car, used computers and did everything else.
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The Amish and Mennonites are very similar in most of their basic religious beliefs. However the way they live and dress etc. is different from a lot of Mennonites. "Mennonite" is a very broad term ranging from those who drive horse and buggy and are very similar to the Amish and those who pretty much blend in with the world.