None, except for the fact that άκουσε (sing., 3rd person) is also the Simple Past form of ακούω. ^.^
Really? No difference at all? I thought that άκουσε με means “listen to what I am about to tell you“, while άκου με means something like “listen to me when I speak”, at least that’s what I understood from the difference between γράφε and γράψε.
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. :)
I think that some verbs or constructions require dative which shows indirect object. Δώσε μου means "give (something) to me", it doesn't mean "give me (to someone)". That would be Δώσε με. Άκουσε με simply means "hear me".
Modern Greek does not have the dative case any more :)
With personal pronouns, standard modern Greek uses the genitive case instead (though in northern Greece, the accusative is often heard); with nouns, usually preposition σε + accusative case.
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.