To be reactive is to be ready to react or respond to something else — as opposed to ready to act on one's own. A person who's reactive does things only in response to others.
To react is to do something in response to something else. When someone pinches you, for example, you react. But if you're a reactive person, then you only react; you're always ready to react but not to act on your own. You're rather lifeless unless something or someone else causes you to do act. This is usually seen as a negative trait in people, unless you're talking about, say, firefighters or cops. We admire people who don't need prodding to get things done. Some chemicals are called reactive, too, because they react readily with other chemicals.