What's the difference between a tune and a melody?
Tune can mean a song as a whole as well as its main musical theme, while melody is usually restricted to the latter, at least nowadays. If someone were to say to me "He played a delightful little melody on the piano", I'd expect him to have used a single finger, while "a delightful little tune" is more likely to have involved both hands. ("A delightful little song" would remove all ambiguity.)
Melody is more likely than tune to be used in opposition to harmony. If you're singing in the choir and ask "who has the melody at the fifth measure", the answer will be (probably) "the sopranos"; but if you ask "who has the tune", you'll either get puzzled looks, or someone will play an A on the piano.
It's a bit subjective, but I think tune is less formal than melody: if you're whistling it, it's a tune, but if Maria Callas sang it, it's a melody.
Mainly spelling. That is, tune and melody are very, very similar; I would treat them as synonyms. The MacOS X dictionary (not a dreadfully authoritative source of information) defines 'tune' as MELODY and 'melody' as TUNE, supporting my contention. There are other words used in the definitions, but they come up very similar:
familiar melodies: TUNE, air, strain, theme, song, refrain, piece of music; informal ditty.
she hummed a cheerful tune: MELODY, air, strain, theme; song, jingle, ditty.
Tune has some non-melodic connotations when it is a verb - tuning an engine is different from whistling a tune.
A "tune" to be a short section of music, usually containing harmony and possibly multiple melodies. Tune can also mean solely the melody, but almost never solely the harmony.
"Being in tune" means that multiple people, when trying to trying to sing/play the same note, have the exact same pitch.
A "melody" is single series of notes that can be hummed, sung, or played by a single person. A melody is not any random series of notes, but it is the most important part of the music that is supported by the harmony. In one way, it is the part of the music that you remember and associate with the music. When a band director says "They [naming a section of instruments] have the melody," that means that the section has the most important part.