A key in the Common Practice Period (about 1500-1900) is more than a scale. There are implied harmonies and differences between notes. Short explanation follows.
A scale is simply a bunch of notes arranged in order by pitch (or frequency.) Over the years (at least from ancient Greek times to today) people have abstracted from musical pieces those pitches which seem important in some way then arranged them in order. Gregorian Chant has several scales (called modes) abstracted from the corpus of chants.
A key is quite a bit more. In a key, some notes are special (also true for modes) and certain chords have special meanings. I'll use the major and minor keys for examples (most other musical organizations aren't generally called keys). A major key ( C-major ) consists of a set of 12 notes, (approximately) evenly spaced. Seen of these are termed "diatonic"; naming them for historical reasons as C, D, E, F, G, A, B (in order of pitch from lowest to highest). These are unevenly spaced in the sense that between E and F and between B and the next C (they are used cyclically) there is no note; between each other pair of notes there is another (named "chromatic") notes; these have two names, but I'll mostly use one name (to make it easy to write without re-editing). C, C#, D, Eb, E, F, F#, G, Ab, A, Bb, B (the b lowers the note above an the # raises the note below.) Each note has a separate musical place to are not really exchangeable. The note C is termed the tonic and G the dominant (historical naming again).
In addition to the notes, combinations of notes played simultaneously are important for key structure. The "tonic chord" is C-E-G; in the key of C-Major, this chord is used to end (almost all) music in the key of C (and it is used to begin a large plurality of such pieces.) I'd also like to note that the causality really goes backwards; pieces that end on a C-Major chord and rhythmically and texturally seem to stop there are called being in C. To really be in the key of C, the chord sequence G-B-D (a G-major chord) followed by a C-major chord is used to end the piece (or to at least mark off sections). Other common patterns like F-A-D (a D-minor chord) followed by a G-major then a C-major chord are common. Pieces in C-major do not end on a d-minor chord for example. One can use chromatic chords without leaving the key of C: D-F#-A followed by G-B-D then C-E-G still sounds like C-major. There are lots more a "French Sixth" (historical naming again) chord has two chromatic notes but if treated properly need not leave the key. F-A-D (D-minor chord in first inversion) followed by Ab-C-D-F# (the French Sixth) then G-C-E-G (cadential tonic 64 which has other names) then G-D-D-F (dominant seventh) ending with C-E-G is common. Two important things should be noted here: names don't mean much, only usage; also the "chromatic" notes also belong to the key. Other keys have the same structure but the base note (or bass note or tonic) id different.
Minor keys are a bit more complex. The diatonic notes are C, D, Eb, F, G, Ab, Bb, however the notes Ab and Bb (scale steps 6 and 7) are mutable; they may occur as A or B without being termed chroatic. The tonic is C-Eb-G (C-minor chord) and the dominant is G-B-D (major chord; mutated step 7). The patterns with Ab-Bb or A-Bb or Ab-B or A-B in either order are common, even in the same phrase.
Some (most) texts write three minor scales called Natural, Melodic, and Harmonic minor scales. There are some common uses (I've posted these elsewhere a few times) but none lead away from the key of C-Minor. Jazz theory treats each of these patterns in special way, a bit differently from the classical theory.
So, scales are ordered collections of notes abstracted from musical practice. Key are large-scale melodic and harmonic structures that tend to be used a lot. It's the music, not the labeling or taxonomy.