The 1/2 HP pump won't pump as much water at the same pressure, and won't provide the same pressure when it is pumping as much water. In general you can expect that the 1/2 HP pump will provide only about 2/3 as much water as the 3/4 HP pump.
There is a wide range of 1/2 HP pumps from many manufacturers, ranging from 5 GPM rating to 15 GPM rating. The high-GPM rated pumps won't deliver as much pressure and the low-GPM rated pumps won't deliver as much flow.
Five minutes of running water before the pump comes on is not surprising if you are running more than about 3 gallons per minute. Not being able to deliver more than 20 psi when you are running water suggests that you have a seriously undersize pump. How many gallons per minute are you getting when the pump is delivering only 20 psi? Do a timed test for by measuring how many seconds it takes to fill a 5-gallon bucket.
Post the make and model number of both pumps so we can get some idea of the characteristics.
Also, post the model number and size of the tank. Is it a bladder tank (with an air connection on the top) or is it a non-bladder tank?
In the meantime, I suggest setting your pressure switch at 30 psi start and 50 psi stop. Then, with all of the water out of the tank and the power off, set the air pressure in the top of the tank at 28 psi. That will give you maximum available drawdown.