Tub spouts typically come in two varieties, the one you have - a compression fitting, and the other, that screws on. Many of each variety are available and work fine. The compression type does, though, require that the pipe be in good shape and the o-rings can't be damaged when installing. A couple of things can compromise the installation - tightening the set screw too much and deforming the pipe or not cleaning off the burrs at the end of the pipe and slicing the o-rings. If the thing is really old, the o-rings could be getting dry and cracking. It is usually pretty easy to see if that is the source of the leak. Put it in the shower mode and watch! Make sure that the shower is hitting the tub and not the walls. This eliminates the walls, but not the pipe up to the showerhead as a source. If you don't get any leaks then, move the showerhead around so it hits the walls and see what happens.
Basically, unless it is plumbing, it is the wall construction. Note, neither the tile nor the grout is totally waterproof, but they significantly protect against liquid water from getting through. They will allow moisture vapor to get through and condense, but that would not cause a stream of water, only enough to rot the studs over time. There should be a lip on the tub to prevent liquid from getting past, but if the tub is not set level and there is any damage, it can leak through there.
Check any caulking at the tub/wall junction, and if compromised, clean it out, let it dry, then recaulk. You should have caulk there, not grout.
A failure of grout should not cause a leak of the nature you are talking about.
www.johnbridge.com is a great place to get tiling help, if you need it.