While the two words mean almost the same thing, here is an implied difference between them. The word "fragment" implies that a smaller piece broke off roughly from a larger whole, while the word "part" simply means it belonged to a larger whole.
For instance, a bicycle part may be a pedal or handlebars. Even if a pedal falls off, it would likely fall off in one entire piece; so we would still say it was a "part" of the bike.
However, if a bicycle were hit by a car, pieces of it may break off roughly: a little piece of chain, a chunk of the seat, a chip of painted metal. These pieces would not be called "parts," because they are not in their original form. These are bike "fragments."
Both "parts" and "fragments" would be considered "pieces." The difference is that "parts" have a recognizable form or amount, while "fragments" do not.