It depends on the size, complexity and number of airports served by the approach control (in the U.S. typically called a TRACON- Terminal Radar Approach Control). Also, there may be different controller positions (sectors) within the TRACON variously (internally) named "Approach," "Arrival," "Finals," "North Arrivals," "East Departures," etc.
Again, these are internal sector designations (they can vary significantly between facilities) and typically are used for internal communication (sector to sector or sector to a specific tower) and communication with other (usually adjoining) facilities for traffic coordination purposes.
Generally, when you have been "handed off" from one air traffic facility to another (like being handed off from an ARTCC - Air Route Traffic Control Center to a TRACON) you will be instructed to contact "XYZ Approach" on a specific frequency. Internally in the TRACON this could be (but not necessarily) identified as the "Arrival" sector. This controller would then be responsible for providing separation, vectors, speed control, altitude assignments, monitoring/controlling your arrival via a Standard Terminal Arrival (STAR) etc. until you get closer to your ultimate destination airport, which could be the Hub airport (major/primary airport) or any one of many possible satellite airports within the TRACON's area of responsibility.
Again, there could be several internal (to the TRACON) other sectors that you could subsequently be handed off to depending on the size, complexity, and number of airports within that TRACON's airspace. Or you may be just talking to one TRACON controller until instructed to contact the tower.
Also, it is common that once you're handed off to that TRACON and instructed to contact "XYZ Approach" on a specific frequency, during the internal handoffs between sectors within that TRACON you are instructed to "contact approach" on (some frequency) as opposed to being instructed, for example, to "contact finals" (or using some other internal TRACON sector name) .
Lastly, an "Arrival" controller and an "Approach" controller, within a TRACON, are just different generalized nomenclatures for controller positions within that facility and their functional roles are not specifically distinct like a Ground controller vs a Local controller in a Tower. And these nomenclatures may differ depending on the TRACON.