好きじゃない is the negative of 好き, so it means "don't like" or "not fond of".
嫌い is stronger, closer to "dislike" and, depending on context, sometimes "hate".
Much like in many other languages (including my own), it's usually more polite to say that you're not really a fan of something rather than directly saying that you hate it.
So something like あまり好きじゃない かも (roughly "I might not really like it" or "I'm not really a fan of it") is a much softer way of expressing dislike than using 嫌い
PS: This kind of "indirectness" certainly isn't unique to Japanese, though you could argue that it's a lot more common there.
