άκου με or άκουσέ με are basically the same thing: 'listen to me'. Like γράφε and γράψε, the difference is that the first implies a more continuous, unspecified duration, like '(be) listen(ing) to me (every time I talk and for however long that takes)', while the second implies a bit more limited duration: 'listen to me this time/ for the x amount of time this is going to take '.
However, these are largely philological grammatical differences and thus mostly theoretical. In practice, 'listen to me' is a very good translation for both phrases. I should add, a) context is always important and could make these differences more pronounced, b) speakers tend to observe grammar rules more closely for some verbs rather than others. But that is the essence of colloquialism of course. :)
μου is for the indirect object, με for the direct object.
The direct object of giving is the thing, the indirect object is the recipient. So Δώσε μου το βιβλίο "Give me the book", "me" is the indirect object, the recipient, so it's in the genitive case in Greek.
Whereas in Άκουσέ με (note double accent if you use the three-syllable long form!), "me" is the direct object, the thing or person heard, so it's in the accusative case in Greek.