What is the difference between a router with an internal switch and a router without?

Its a matter of convenience.

If you already have a switch in place, where all your devices are connected, then all you need as the router. You would then connect that router to the switch, thereby giving all those devices connectivity.

However, if you do not have a switch, a router with one integrated, saves you the need to buy a separate one, since everything would be connected to the switch portion of the router/switch combo.