Is there any difference in sound between mě and mne? If not, shouldn't both be accepted in dictation?
Not really. N is alveolar so itong is just behind the teeth, ň on the other hand is palatal, so tongue is on hard palate...
Anyway, I avoided the terminology but indeed made a mistake in the process. I do not agree alveolar is "just behind the teeth", it is a bit father, Czech N is not, unlike some other languages) dental or dento-alveolar.
Ň for me comes pretty far on the hard palate, but perhaps not all the way to the soft one, but be aware it is the farthest point of contact that counts.