This was a Type What You Hear exercise. My response -- "Nemá mne ráda" -- was marked wrong because I used "mne" and not "mě." Earlier today, in a similar exercise, my response was wrong because I used ""mě" instead of "mne." I posted a comment on the first instance, too; I'd really like to know when (if) one should be used instead of the other. Can someone explain, please? Or make both acceptable? :-)
We cannot make both acceptable. It is technically not possible on our end.
That said thou, in listening exercise the robot does say MĚ. There is a difference in how you pronounce the two. If it was an exercise to translate from English to czech, both MĚ and MNE would be acceptable.
I'm not a native speaker, but from what I understand, in Czech basically every letter is pronounced, which is not necessarily the case in English. So, in your examples, I would expect that there would be a bit more "M" heard in the second than in the first -- like two "Ms" in a row, which actually there are. In everyday life, though, what you hear would probably depend to some extent on the quality of the speaker's pronunciation. Playing the exercise on slow speed may help. And someone else may be able to tell you more.
They are equivalent in their ability to bear stress. Both are inconstant clitics, so they can appear as clitics and participate in clitic clusters in accusative/genitive pronoun positions. Or they can act as regular words. This is very different from the dative pair "mi" (constant clitic) and "mně" (inconstant clitic).