I actually had to look up what the difference between a key and a scale is, as I thought the terms were pretty much interchangeable. I've edited the last part of my other comment to reflect this:

A scale is actually an ordered set of notes belonging to a key. A key is just an unordered collection of notes. I got this wrong earlier.

So playing all the notes belonging to C major in ascending order is playing a scale, and playing the notes in any order is playing in the key of C.