Thank you. Now I found out everything what I wanted. DekujiHi Ostin, the singular personal pronouns (já, ty, on, ona, ono) have more than one form in the same grammatical case, and in respect of já, they're shown in the table here (prirucka.ujc.cas.cz), so mě (or mne) can be either 2. pád (genitive) or 4. pád (accusative), mně can be either 3. pád (dative) or 6. pád (locative/prepositional). This grammar phenomenon (more than one form in the same case for the personal pronoun já) doesn't exist in standard modern Russian, which is possibly why you're asking the question. Mě and mně are pronounced the same - [mɲɛ]. Mne is pronounced as written - [mnɛ].
Grammars usually describe these alternative forms of the pronoun as:
1) "long" form or "short" form. This means only that mne is longer than mě, nothing more.
2) Used with a preposition, or not used with a preposition
3) Emphatic or non-emphatic
4) Neutral style or "bookish"/formal style.
Looking at the Czech table again, for the genitive and accusative cases, you can see:
Neboj se mě/mne. Don't be afraid of me! You can use either form. Bát se takes the genitive (koho/čeho).
U mě/mne se nic neztratí. Nothing gets lost with me, nothing of mine gets lost. You can use either form. U takes the genitive (koho/čeho).
Mě/mne jsi nečekal. I'm not the person you were expecting / waiting for. (You can use either form. Mě/mne are in the accusative, čekat koho/co).
The Czech page says the two options (mě, mne) for the genitive and accusative are effectively interchangeable and can both be used as enclitics (second element in the sentence word order), both can also be used with prepositions, both can also be used for emphasis. "The form mne is not used conversationally, it sounds bookish." You may also find this page "Pronouns" helpful (cokdybysme.net).
But your question is vague. The thread title says mne and the text says mně - two different words. In the genitive and accusative there's effectively no difference between mě and mne - they're interchangeable in almost any syntax situation. In the dative (mi/mně) there is a significant difference, but you didn't ask about that.
If you have a more specific question about the usage of mě/mne in a specific context, please give the full sentence and say what you're not sure about.
As for the vocative, that's a different kettle of fish (to je jiná kapitola, to je jiný kafe, это из другой оперы), so you'll have to open a new thread for that, as this forum only allows one subject question per thread.