I know that the plane must have two autopilots for flying approaches, but why have individual buttons for each autopilot? Wouldn't it be easier to have both autopilots work at the same time for the entire flight?
Just like real-life pilots, you only want one flying the airplane. The other is running in the background and can take control when selected. When you select which A/P is active, that just controls which A/P is actually flying the a/c.
Depending on the system architecture, how monitoring and fault detection are designed, the system can have one or more reversion modes that can be implemented. These can be automatic or require crew action, depending on the system and phase of flight. For example, in cruise you could fly on one A/P when the other is failed. But in approach, both A/P would need to be active to fly a coupled approach.
Actually many planes have 3 autopilots. One actually flies the plane and others check the one flying the plane. with 3 autopilots if the results from one, differs from the other two, it gets shut down and the plane continues with two. If there are only two autopilots and one gets different results from the other, both will shutdown and the pilot has to resume hand flying the plane.