What's actually happening
Your upstairs tub spout is leaking and the downstairs one is fine. in most cases this is a worn part inside that specific faucet, not anything wrong with your plumbing as a whole.
The two usual suspects are a worn valve cartridge (drips from the spout when the handle's off) and a tired spout diverter (dribbles from the spout while the shower runs). Both are cheap parts.
The reason this page treats an upstairs leak differently is simple physics. Water that escapes behind the spout upstairs has nowhere to go but down, straight into the ceiling of the room below.
So before you buy anything, figure out whether the water is coming from the spout tip or from behind the spout at the wall. That one answer decides whether this is a lazy weekend fix or something you handle this week.
The likely causes, ranked
Start with the cartridge or washers. This is the most common cause of a spout that drips when everything's shut off. The internal seals harden and crack with age, and water squeezes past.
Next is the diverter. If your spout has a little lift-up gate that sends water to the shower, that mechanism wears out. When it can't seal, water leaks from the spout instead of going up to the showerhead.
Third is the spout connection itself. A loose or poorly sealed spout lets water escape behind it, and on a second floor that water runs down the pipe into the ceiling. You might see almost nothing at the tub.
Less common but worth knowing: a cracked spout body or a corroded pipe nipple. Old brass and galvanized pipe rots from the inside and can split, leaking only under full pressure.
On two-handle faucets, worn valve seats are the quiet culprit. You replace a washer, it works for a few weeks, then drips again because the pitted brass seat keeps chewing through the new washer.