Independent clauses can be connected (or separated, depending on your point of view) in a variety of ways. When two ideas come together and either one of them can stand by itself — as its own, independent sentence — the following kinds of punctuation are possible. (Review, also, the sections on Coherence: Transitions Between Ideas and on avoiding Run-on Sentences.)

1Period + start a new sentence

My grandmother refuses to go to bed early. She thinks she's going to miss out on some of the action.

2 Comma + a cute little conjunction (and, but, for, nor, yet, or, so)

My grandmother refuses to go to bed early, and I'm afraid she's going to catch a bad cold.

3 Semicolon by itself. Where you have used a semicolon, you could have used a period, but the semicolon, you felt, is better (probably because the independent clauses are so closely related and nicely balanced).

In spite of her cold, my grandmother refuses to go to bed early; she is afraid she will miss something.

4Semicolon + big ugly conjunction or other transitional expression
(however, moreover, nevertheless, therefore, as a result, consequently . . . )
followed by a comma.

My grandmother has stayed up late four nights in a row; as a result, she cannot seem to get well.