In common usage, a phrase is usually a group of words with some special idiomatic meaning or other significance, such as "kick the bucket or all rights reserved.
As a sentence: She was going to kick the bucket.
Fragment: hit a nerve.
As a sentence: The dentist hit a nerve.
What the difference fragment/phrase.
A fragment is a piece of something.
So the term "fragment" is applied to the phrase from the point of view of that complete something. The phrase, when it is considered to be only a part of that complete thing, is called a "fragment" and that complete thing is called a sentence.
A fragment can contain more than one phrase.
Your question is like asking, "What is the difference between a clay shard and a clay tile?"
I reserve the term 'fragment' for a sentence which is grammatically incomplete. Like this one. I use 'phrase' for a group of words which conveys a complete concept but falls short of the requirements to be called a 'clause'. 'Clauses' require at least a subject and a verb and many would form a complete sentence on their own, which a phrase can never do. To confuse matters, when discussing grammar I use 'noun phrase' and 'verb phrase' to describe anything which may serve the function of a noun or verb within a larger sentence. They can be only a single word, or a group of words, all the way up to constructions which could form valid sentences containing multiple clauses!
Sadly, one of the hardest things about learning English grammar is there is no agreed glossary of terms, and many authorities use different terms to describe the same thing. :(
