What is the difference between a pump tank and a pressure tank?

Thanks Mike

Probably the pump tank you are referring is one of the small pressure tanks (just a couple of gallons or so) used on booster pumps. A pressure tank is larger and not directly mounted on the pump. At any rate, that's my best guess on it.

This is an above ground well pump with a small metal tank sitting next to it and has an air valve on the top of the tank for adding air.
It is full of water and pump is short cycling badly the tank is very old so I want to replace it
I went online to look at tanks and saw two types being sold a pressure tank or a pump tank both look the same to me in the pictures so I wasn't sure which tank I should buy.

A pump or bladder tank, (see image) is a smaller tank with a membrane
Your bladder tank membrane is ruptured. Replace it!
A pressure tank is a larger galvanized tank,(see image) that uses trapped air as a cushion in place of a menbrane,
They both do the same thing and that is to prevent the pump from cycling on and off every tine a valve's opened. Good luck, Tom

There are two types of well tanks, pressure tanks and bladder tanks. They are actually both pressure tanks but one has a bladder and the other does not. We just don't have another name for the non-bladder tank.

Pressure tanks (non-bladder type) are usually galvanized. Bladder tanks are usually painted. Galvanized pressure tanks are galvanized inside and out. Bladder tanks have the scharder air valve for pressurizing the tank.

Apparently you bladder has ruptured and the tank has become water logged. One of the purposes of the bladder is to prevent this.

As a temporary measure you can empty the tank of water and pressurize the tank to 2 lbs below the pump cut in pressure to stop the pump short cycling. Some times you get an odor to the water because the inside of the tank is not coated for use like this. However, you have water until you can replace it.