No, they are not synonymous.
By definition, an independent clause can be a standalone sentence, but it doesn't have to be.
It is raining, so I won't go out for dinner.
This is a sentence composed of two independent clauses joined by a coordinator "so". In the same manner independent clauses connected with semicolons also form one sentence with multiple clauses.
Also a sentence doesn't have to comprise independent clauses. Incomplete sentences, which often appear in literary works, are also sentences in their own right. Imperative sentences usually lack necessary components to independent clauses. In conclusion, "sentence" and "independent clause" are two different yet related and often confused concepts.