Having a separate process can be useful if there are components of your application that do not necessarily need to both be running to be useful to the user, and the background task is critical to application "correctness" (either now or in the future). The classic example of this is an app that has a service where the service saves or uploads some data that is critical to your application (critical meaning the only way to get the data back is to have the user re-enter it!). The service might be responsible for doing something like uploading or saving data, while the activity is just the interface for the user. So developers should decouple these two components to prevent problems that may arise from my next point..
Android was designed to run in a resource (especially memory) constrained environment, so processes deemed unimportant are killed periodically to open up memory for important ones by the "low memory killer" (LMK) (if you Google this you'll get tons of information on the topic). Things like foreground processes are understandably given a higher priority since they're currently in use, but they're sometimes killed off as well for reasons like consuming too much memory. Now, imagine you need to save off some data to a database after the user does something in the app and you use a service to do so to ensure that it is done even if the user navigates away from the app. Unless you create the service in its own process the process containing both the activity and the service is likely to be killed since the process belongs to a non-foreground activity.
However it is not always necessary to place the service in its own process, oftentimes simply giving the service its own thread will suffice; it's very application specific. I would only place a service in its own process if it took longer than maybe a few seconds (long enough for the user to navigate away from my application and for the LMK to step in) to perform some task in the background and that task related to the "correctness" of my application (I.E. saving data for later). For something like caching, stick to threads, since if the process gets prematurely killed you can just recreate that data later.
Another reason to have a separate process is if you're running a global service (a service that can be used by applications other than your own) that maybe you provide an interface with via an Activity for configuration.
As for the performance question, there will definitely be a performance hit for something like this. Interprocess communication is not cheap, so you should really only use a separate process if you fit into a specific use case, like the ones mentioned above. Also, there's a certain amount of memory overhead for maintaining a process, so that's another performance hit.