If referring to phonetics then thee would be for emphasis to a construed subordinate with thuh used generally. Really idiosyncratic.
If referring to written words then use thee if you find yourself in Pennsylvania Dutch country trying to hail a horse drawn taxi or thuh if trying to play dumb while text messaging.
True. The above was just for fun. But colloquial English these days behaves as if rules are to be made up as you go along. EG, locally I hear 'I'm going to thee Applebees Bar & Grill' as often as 'I'm going to thuh', but then again there isn't an abundance of Rhodes scholars in this neck of the woods.
If referring to written words then use thee if you find yourself in Pennsylvania Dutch country trying to hail a horse drawn taxi or thuh if trying to play dumb while text messaging.