But if you roll along with the car off and there is no noise then it cant be tyres.
If its only there when the engine is on then its going to have to be something inside the gearbox/diff. As to what it is.... I cant help you.
The fact that it doesn't make a noise when idling is annoying, that could point to it being an input shaft bearing on the gearbox or something along those lines.
Being that it makes the noise when moving in neutral, but not when sitting there idling in neutral doesn't seem to make a lot of sense. From that you would have to assuming the diff but then it should be making it while you're rolling with the engine off
Get a new gearbox and see if it goes away