Apart from the use of bellows and reeds to make the sound, like chalk and cheese. A Melodeon is like a button accordion, will give a full and slightly chunky sound, and is often restricted in which musical keys you can play on it. The concertina is usually a much smaller six sided instrument (there are other variations) with a much lighter sound, often fully chromatic. Each button creating one note rather than the melodeon which produces 2, depending on whether the bellows are pushed or pulled.
That is a VERY general description. There are loads of different types of both, worth taking the time to look into.
Page of concertina pics: My link
Page of melodeons: My link