Lets assume your cooling system is completely full, with no overflow. What would happen?

As the engine heats up, the coolant heats up. The thermostat opens, coolant begins circulating through the radiator from the block. Now all the coolant is hot, what happens? Pressure builds up.

With a 16 lb rated radiator cap, the pressure can get to 16 lbs before the cap opens. This allows the system to be pressurized, allowing a higher temperature. During normal driving, this probably won't even be needed.

But then lets say you run up to 80mph for a few minutes, and then come up on the Disney exit. Traffic. Now you're crawling along at 10mph. The engine is hot, and that radiator just isn't cooling enough. Increased heat, increased pressure. It reaches 16lbs, and the cap opens. Coolant gushes out. Pressure is relieved, cap closes. BUT, now there will be air in the system. As much air as coolant was lost.

What if there's an overflow tank? The coolant gushes out, into a container. BUT, instead of sucking air back in when the pressure is relieved, coolant will be pulled in. That's all the overflow system does, relieves the system of pressure, into a container, so the coolant isn't lost, but rather pulled back into the system when it cools.

If the cap doesn't open, what happens? Well...the cap is the relief valve. It's the only one. Without it, the increasing temperature would increase pressure in the system. Then you'd be testing the strength of various seals, gaskets, and hoses. That can't be good.