Learn French: Le loyer est cher ; par ailleurs, le quartier est bruyant. - The rent is expensive; furthermore, the area is noisy.

They belong to the same family of argumentative markers, but they aren't strictly interchangeable. D'une part... d'autre part (on the one hand... on the other hand) explicitly sets up two parallel or contrasting points. Par ailleurs (furthermore / besides) simply adds a new, separate piece of information to support your overall point. Unlike d'autre part, it stands alone and doesn't require a "first part" to have been announced.

Does par ailleurs have the exact same nuance as en plus?

They both add information, but par ailleurs is a bit more formal and structured. While en plus just stacks another fact on top, par ailleurs (which literally means "by elsewhere") signals that you are bringing in a different aspect or category of the situation. Here, the first point is about money, and the second is about the environment.

Is loyer related to the verb for renting?

Exactly! The verb louer means both "to rent" (as a tenant) and "to rent out" (as a landlord). Le loyer is the noun specifically for the rent money you pay.

Why use a semicolon before par ailleurs instead of just starting a new sentence?