Many a thread we have on this...mmmmmm?
Well they are all still clones which GL said himself. They just changed names. And also the Empire is supposed to have a sort of Nazi pun so that another reason they are called that.
They are the backbone of the empire and are the most commonly seen troops throughout the galaxy.
"Stormtroopers" fits quite well since they enforcing the will of the empire and beforehand during the clone wars they were fighting for the good of the galaxy.
Search please, search.
First off, the name change is nothing big. Like Echuu said they no longer needed to go by Clonetroopers and Stormtroopers is a much better way of describing a terrible Empire.
The armor, well, the Clone Wars was a big elaborate war and they were the flashy new warriors of the Republic. With the change to the Empire they no longer had to impress anybody and there was no real war going on until the OT.