What could be some reasons why a tub faucet's spout leaks when water is turned off? E.g. some gasket within the mixing valve perhaps? The shower doesn't leak, just the tub spout.
If a new gasket doesn't clear up a drip, the valve seat may need to be resurfaced or replaced. The big box stores have replacement seats and the wrench to do it, and may have the tool to resurface. If the correct replacement seat is available, that's the way to go IMO.
All such tub spouts have a sealing washer or cartridge with a sealing gasket. You need to unscrew the unit's cap and withdraw the central core.( with main water supply turned OFF ) Now take this to your plumbing store and get a repair kit. They are available fairly cheaply for almost every unit.