The term Subject describes a particular job in a sentence. The term noun phrase describes a phrase headed by a noun (in other words the phrase is built around a noun). The first word, Subject tells us how a phrase relates to different parts of the sentence. The second term noun phrase tells us about how that phrase is built. So a phrase can be both a Subject and a noun phrase. But the terms Subject and noun phrase mean different things. Not all Subjects are noun phrases, and not all noun phrases are Subjects!
The term Predicate can be used in different ways. But when we talk about syntax/grammar the term Predicate describes a particular job in the sentence. The term verb phrase tells us that a phrase is headed by a verb.
The Predicate in a sentence is that part of the sentence that (usually) comes after the Subject. So in:
The Predicate is walks slowly. We can ask what kind of word or phrase the Predicate is. The answer is it's a verb phrase. In English, the Predicate in a sentence is always a verb phrase. BUT! Not all verb phrases are Predicates. In the phrase a talking dog, the item talking is a verb phrase, but it is not a Predicate.
In this post I have the names of the different jobs that a phrase can do with a capital letter. These jobs are called grammatical relations or syntactic functions.