Nothing really, it's mostly just a similar name for the same thing. But while a booster could quite happily be called an amplifier, an amplifier can't always be called a booster (if that makes sense?).
A 'booster' usually refers to an aerial amplifier, so if a customer asks for a 'TV booster' or 'TV amplifier' I know exactly what he means.
But is someone asked for an 'audio booster', it's not a term that's used, it should be 'audio amplifier'.