In the notes section of personal pronouns, it states that mě and mne are pronounced the same, but listening exercises will not accept them in place of each other. What is the truth?
That is not correct.
MĚ and MNĚ ar both pronounced /mňe/ (soft)
MNE is pronounced /mne/ (hard)
Diacritics is very important here.
Vlada Fu describes it perfectly. A notice: we don't use "mne" in common language, it's very bookish. We use only "mně" and "mě". And the sound is same for both of them.