A leaking spout is an annoyance, leads to stains in the bathtub and wastes water as your usage increases from the constant dripping. Unlike faucets that are just for bathtubs, bathtub and shower faucet combinations leak out of the spout because of one of two possible worn out parts.

...
Leaking from the shower and tub spout comes from one of two possible sources.

Worn Diverter

A bathtub and shower combination spout will leak water while you are taking a shower if the diverter has worn out. The diverter is the small bar or handle you use to change where the faucet's water flows out. Lifting the diverter up will cause the water to come out of the shower head, while pushing it down will cause the water to come out of the spout. The diverter wears out over time and must be replaced for you to continue taking showers.

Replacing the Diverter

Since the diverter is located inside the faucet's spout, you must replace the entire spout once the diverter starts to wear out. You first must determine how the old spout is attached to the water pipe by looking at the underside of the spout. If you see a screw or a small hole where a screw can fit, you must loosen or remove the screw with an Allen wrench before you can remove the spout. If you do not see a screw, you must turn the spout counterclockwise to unthread it from the water pipe. Use a pipe wrench to turn the spout if you cannot do it by hand.

Worn Faucet Valve

Different types of faucets use different types of valves, but all valves control how much water flows through the faucet. You actuate these valves each time you use one or both of the faucet's handles. Like a diverter, the faucet's valves will wear out over time and from use to the point they can no longer keep the water from passing through at any time.

Replacing Faucet Valve

With any type of valve setup, you must first turn off the water in the house at the main water valve before you start to take the faucet apart. Turn on another faucet to make sure the water pressure in the house is gone. Remove the retaining screw that holds the faucet handle in place, which either will be hidden under a cap in the middle of the handle or in a small hole around the base of the handle. Remove any other pieces of the faucet to exposed the valve, including metal sleeves, nuts and pins. With cartridge valves, you must pull the cartridge out and insert a replacement. Ball assemblies require you to remove the ball and rod, as well as the springs and seats. Replace the springs and seats, coating them in plumber's grease. Valve stems require you to turn the stem counterclockwise to unthread it from the faucet. You can either remove the different seals from the stem by removing the retaining screw at one end, or you can replace the entire valve stem.