One has a subject and a verb; the other doesn't.
A clause has both a subject and a verb. There are two types of clauses: independent and dependent. Independent clauses can stand alone, whereas dependent clauses cannot. In the sentence I just wrote, the first part was an independent clause, and the second part was a dependent clause because it had the subordinating conjunction "whereas" before it.
A phrase, on the other hand, has a noun or a verb, but not both. This phrase, for example. It had a noun, but no verb.