Me llamo is the natural way to say it in Spamish. Mi nombre es is becoming more popular becuase of the influence of English grammar. Nobody would say "Mi nombre es" in a regular conversation with friends or acquaintances.

Mi llamo would be the native thing to say. Mi nombre es = sounds too much like a translation.

Both are ok for any context. I think in English is more common "My name is" but in Spanish both are used often. e.g. Hola me llamo Israel y soy Australiano Hola mi nombre es Israel y soy Australiano.

Hi, you can use both, it´s the same. Bye

Es lo mismo.