In this context 'exert' is a slightly stronger verb than 'exercise'. Although both have physical meanings as well, here it is talking about use of power and political influence. To exercise your powers is to use them in the normal way they are used. A judge exercises his powers at a trial. Exercising influence is therefore more persuasive than anything. If you exert power or influence, it is generally more compelling, and possibly more threatening than just being persuasive.

No, in this context they mean the same thing. Normally however, "exert" means, "to apply" "to bring to to bear" "to make use of" or "to employ." "Exercise" normally refers to an activity requiring physical effort engaged in for the purposes of keeping fit and healthy.