As an example: "That noise worries me" implies that you think the noise indicates a problem or that something bad might happen. "That noise bothers me" might mean that you're worried about the noise, or it could just mean that the noise is just loud, distracting, irritating, etc.

If something is bothering you, it could mean worry, but it could also mean something is annoying you, or interrupting you. It is a word with more meanings. To say something worries you mean just that. It is making you stressed and worried. I hope that helps!