The overflow is there because when coolant gets hot, it expands, and will run into the reservoir at 16psi (cap pressure). As it cools, it will suck the coolant back in the radiator. No air.
If you don't run an overflow, excess coolant will just dump out. Air will then replace it when it cools. Air in the system will cause overheating, which causes more coolant dumped out (as temperature goes up so does pressure) and more air in the system. Eventually you will overheat. May not be right away, but you will.