Aren't Mennonites people who follow the religion of The Amish? They are called Pennsylvania Dutch but in reality are descended from German immigrants.
No, they are not the same. Somewhat related, but far from identical. Both I believe had their roots in the pacifistic Anabaptist movement of renaissance Germany, but there was a great schism at some point. One half became the Amish or the predecessors to such, and the others became Mennonites.
There was a leader... Theodore Munser or something like that was his name... who led one significant faction while it was still in Germany. He led an extremely violent, jihadist sect of Anabaptism that I think, but am not totally sure, became incorporated into the Amish tradition. Amish came to colonial Pennsylvania in the early 1700s, and yes, as you pointed out, are indeed ethnic Germans. Mennonites followed a man named Menno Simon (or, again, something like that) who was more pacifist. Most Mennonites moved to Russia in the 1700s to avoid conscription by the German/Prussian armies, which was anathema to their belief system. Flash forward 150 years or so till when the czars were launching pogroms against the Mennonite colonists in Russia (which of course spoke German and were pretty socially distinct) much as they were Jews. Many fled for Canada and various parts of Latin America. Those that did not flee then gradually returned to Germany from Soviet Russia over the decades.
Because the Mennonite colonists were separate from their ancestral German homeland for hundreds of years, they speak a dialect now (Plautdietsch) far distinct from modern high German--although many know both equally fluently.
The practical difference is that Amish generally are more extreme in rejecting all trappings of modern life and they practice a rather unusual folk religion (complete with the casting of magic spells and the like). Mennonite Brethren are closer to "mainstream" Christians except for their great pacifism, more like Quakers.