Then there should be two different tensions which balance the two weights of the monkeys.
If the monkeys are of different weights, the ropes are not balanced at all (unless the friction in the ceiling is great enough). This is the case, when the monkeys (and, of course, the rope with them) move with some acceleration. Because the rope does not extend during the process, the differences between tensions for each little segment of the rope are the same. Thus, the tension is uniform throughout the rope.
P.s.: yes, I do write "each little segment of the rope" by which I mean a segment of mass dm. You might say, that the rope is massless, but in reality, it does not have zero mass, it has a mass, which is negligible by the conditions of current problem. So this interpretation of breaking the rope in little sections is right.