I think I may be oversimplifying, but I think of a scale being the notes (pitches) you are playing and a key being the chords you play along with it. The chords typically will be the same letters as the notes in the scale, but with an added (major, minor, diminished, etc).
For example -
The notes in the A minor SCALE are: A B C D E F G
The chords in the KEY of A minor are: A minor, B diminished, C Major, D minor, E minor, F Major, and G Major
I hope that makes sense.