Why does water still come out of tub spout while shower is running and how do I fix it

You are not saying what kind of diverter you have. It is either a button on top of the spout or you have 3rd valve installed on the wall. But regardless, it is not sealing properly. You will have to replace washer.

If you pull up a plunger to divert the water to the shower then more than likely the best way to stop it is to change the spout to a new one.