Tone is what a voice or sound is like. (It answers the question, "What are it's qualities?") For example, "He speaks with a deep tone.".
A tune is like music. It is a group of sounds we like to listen to. A popular game in the US, in the past, was called "Name That Tune". In it you guess which song is being played. You could also say, "I really like that tune."
I hope that helps!