To disturb means to interfere with something that is at rest or at peace. For example, throwing a stone into a calm lake would disturb the surface of the water.
To bother specifically means to annoy somebody.
Imagine you are trying to study and your kids keep coming into your room and making noise. In this case, you could use either one: they are bothering you (because you are annoyed) or they are disturbing you (because they are interfering with your concentration).
Now imagine you are exercising in the park, and there are many insects. The insects are bothering you (because you are annoyed) but they are not disturbing you (because you are not at rest; you are moving while you exercise).
Finally, imagine that you learn that your best friend has committed a violent crime. This would disturb you (because it interferes with your peace of mind) but you probably wouldn't say it bothers you (because it's not annoying).
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