In certain contexts, the difference is negligible. However, whilst they overlap in meaning, they do not match exactly. Bear has many more meanings than tolerate.
Bear means to carry (he bore a heavy burden), so when someone is bearing pain, it means they are 'carrying' it without complaining; they are taking the pain, and not collapsing underneath it. We would not usually use bear of an irritation, or a tiresome person - only of pain or suffering.
Tolerate means to not react to things you find irritating or painful. In my opinion, it is more likely to be used of more minor things - like an annoying co-worker or something. If you do not get angry at the co-worker, then you are tolerant. Tolerate can also mean something calmer, that you are hardly even angered/irritated/pained at all.