From my teacher friend:

A sentence with "Both (one person)and (another person)" or "Neither (one person)nor (another person)" forms something almost like a set phrase, eg 'both patient and doctor', 'neither husband nor wife', 'both teacher and student', where leaving out 'the' is a normal part of the construction, although it is not incorrect to keep it. So it is just a small exception to the normal rules of using 'the'.

I changed some of his words, but this is what he was saying. Does that help?