Both the factory and the service result in singleton objects which areĀ able to be configured by providers and injected into controllers and run blocks. From the point of view of the injectee, there is absolutely no difference whether the object came from a factory or a service.
So, when to use a factory, and when to use a service? It boils down to your coding preference, and nothing else. If you like the modular JS pattern then go for the factory. If you like the constructor function ("class") style then go for the service. Note that both styles support private members.
The advantage of the service might be that it's more intuitive fromĀ the OOP point of view: create a "class", and, in conjunction with a provider, reuse the same code across modules, and vary the behavior of the instantiated objects simply by supplying different parameters to the constructor in a config block.