Dime is the informal imperative,one that is recommended to be used when talking with friends,for example and dígame is the formal imperative used when talking to people we don't know or who are older than us.

Thank you all very much. It is clear now.

Both are imperative forms of the verb "decir". One corresponds to the pronoun "tú", the other to "usted". Hence, one is supposed to be used with people you already know ("dime"). The other ("dígame") is meant for people you don't know or people who rank higher than you. If talking to your bestie, boyfriend, brother or little cousin, use "dime". If talking to the store clerk, a random lad passing by, your local priest, your university professor or a member of parliament, use "dígame".

It's very simple. "Dígame" is only a polite way to express "dime". But are equals. That's it. Saludos.