The difference between
You will not be allowed to speak during the exam.
You aren't allowed to speak during the exam.
is that the first is slightly ambiguous: it may mean that only you won't be permitted to speak or that only this time will you not be allowed.
The second sentence, however, means that no one is ever allowed to speak during the exam: it's in the simple present and indicates a habitual state of affairs.
This is a very technical semantic point about why one tense and aspect is used instead of another. In everyday speech and writing, however, most native Anglophones would not make a distinction. It's really unnecessary unless the person being told that they won't be allowed to speak during the exam asks why not. The answer to that will have to be no one is ever allowed to speak during the exam, which means the speaker or writer should have used the habitual present instead of the future with will not be allowed.
Both rewrites mean They won't let you speak during the exam, but that sentence is also ambiguous. It's a stupid question and an arbitrary answer. However, it gives you a clue about how the test makers think -- or fail to think -- as the case may be.