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I feel like I shouldn't be asking this... ugh.
Every explanation I've encountered left me feeling like a key is the same as a scale. I've been using the word "key" to mean the primary scale the song is in or I'll say a song is in a Major key as long as it is in one of the major modes (accounting for borrowed chords).
A key and its scale are very related, so it's easy to get confused. A key is defined by two things:
The "main" note. You'll hear it called the "center of gravity" or "tonal center" or "home" note. It's the note to which a melody wants to resolve. The proper name for this note is the tonic.
The relationship of the tonic to the other notes. There are many such sets of relationships, but two have the distinction of being called keys: major and minor. (For an in-depth explanation of why keys and modes are not the same, see this question/answer.)
Each key has an associated scale and they share the same name. A scale, however, is a specific, ordered collection of notes which begins and ends with the tonic. It's strict and formal. It exhibits the tonic and the other notes in their formal relationships to one another.
If a piece of music has harmony, you can identify the key (assuming there is one) by the harmony as well. In this case, there will be a tonic chord, the chord to which the piece resolves. The tonic chord also shares its name with the key and the associated scale.
When a piece of music is in a particular key, it isn't restricted to the notes of the associated scale. While you can and should expect the notes of the scale to occur more often than those not in the scale, this is an important distinction that may help you understand the difference. Even if a piece of music strays a bit from the notes of the associated scale, it can remain in the same key so long as the piece still resolves to the tonic note/chord.
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.
You've got some really good answers so far, but I thought I might give some additional perspective...
"Key" is kind of a fuzzy thing. It depends partly on the context. You're not wrong in thinking that key is synonymous with scale, because that's one common usage. In my experience, when someone uses the term "key" they might be talking about any of the following:
Any of those three are the most common uses. Some uses have a bit of cross-over, as in "Blues in A". That tells you the chords are in A, but the scale is most likely a blues scale - and you'll have lots of naturals on the C, F and G notes.
Saying a piece is in a key doesn't necessarily tie you to a scale - a Concerto in D probably has a fairly lengthy chunk that has G# notes, because the modulations aren't taken into account when you name the key of a piece. In jazz, which often modulates frequently, you may hear people talk about the "key of the moment" - that refers to the tonal center of a specific set of chords within a piece that's in a different key signature. Giant Steps is in B major, but it has a couple measures in G, then a bit in Eb, then back to B, and so on.
And you'll sometimes hear singers say "that's my key" when they really mean "if you play it in that key it's comfortable for my range".
None of those interpretations are wrong, even though what each describes is different. I wouldn't stress to much over an exact definition - just try to get your head around what the other person actually means when they say "key".
