A clause is the basic unit of grammar. A clause must contain a verb. Typically a clause is made up of a subject, a verb phrase and, sometimes, a complement:
I didn’t sleep well last night.
Are you listening to the radio?
A sentence is a unit of grammar. It must contain at least one main clause. It can contain more than one clause. In writing, a sentence typically begins with a capital letter and ends with a full stop:
I looked at her and she smiled at me. (two main clauses connected by and)
We didn’t go to the show because there weren’t any tickets left. (a main clause and a subordinate clause connected by because)
In everyday speaking, it is often difficult to identify sentences. We speak in small stretches of language, sometimes just single words or phrases. We don’t always speak in complete sentences, and we often complete each other’s ‘sentences’:
What are those flowers?