Grammarians are looking for categories--for sets of forms that work together. We ask questions such as: How do words combine into "chunks" of various sorts to be used in sentences? How are the kinds of "chunks" that we see in use different from each other? How many kinds of "chunks" can we observe in the data that we collect?
Phrase and clause are two basic categories and terms used by grammarians and linguists. On the whole, phrase is used to categorize combinations of words that do not have the internal structure of a sentence--no subject or predicate is involved in a phrase. Phrases include combinations such as noun phrases, verb phrases, adverb phrases, adjective phrases, and prepositional phrases.
On the whole, clause is used to categorize combinations of words that are sentence-like. A clause has a subject (usually) and a predicate (or we can re-analyze what is given to know that some part of the predicate has been left out to reduce the form). For example, a simple sentence is an independent clause: I like sociology. Or a relative clause is a clause because it started life as a sentence and has been transformed to make a clause that is attached to a noun: I like the sociology course that I took last semester. Or a reduced relative clause is a clause because we know what the full form would be: I like the anthropology course focused on health issues in refugee communities. We know that the "complete" relative clause is that is focused on health issues in refugee communities--and that that complete relative clause can be analyzed as a complete sentence: The anthropology course is focused on health issues in refugee communities.
In sum, a clause is sentence-like with something like a subject and something like a predicate. A phrase doesn't have sentence-like features.
Unfortunately, the definitions for the two are not consistently used and eventually will lead us to problems in other areas. What do we call an infinitive--such as to study sociology or to teach English? Is it a phrase? Is it a clause? Many ESL/EFL texts prefer to call infinitives phrases because they are so different from sentences and other more complete clauses. Many linguistics and grammar studies prefer to call infinitives clauses because an infinitive has a verb--and thus a predicate--and can have a subject. I want you to study sociology. The you is the subject of the infinitive--you could re-analyze the infinitive into a complete sentence: You will study sociology or You study sociology.
That problem, however, lurks
down the road. We can start with some definitions and examples that
are clear cut and generally agreed upon.
Please send me your questions and comments at firstname.lastname@example.org. Thanks.