I can't seem to find a straight definition for both.
A "scale", technically defined, is a sequence of ascending or descending "unit pitches" that form a palette of notes that can be used to form a melody. Most scales in Western music conform to a particular "key"; that is, a sequence of notes that will be "sharp" or "flat" by default. Not all scales have keys; the chromatic scale is a scale of all possible semitone steps used in Western music, while the whole-tone scale is a scale composed of intervals two semitones apart.
Within a particular key, there are 7 notes in a single octave, before reaching the 8th note which is named the same as the first note and is at double the frequency. The seven notes have differing intervals between adjacent notes; sometimes it's one half-step (semitone), while other times it's a whole step (two semitones). The pattern of whole-step/half-step intervals that determine the notes of a key, starting from the note for while the key is named, is whole-whole-half-whole-whole-whole-half. Within a single key, each of those seven notes could be used as the "base" note of an ascending sequence. Each such sequence, created by starting on a different note in the key, is a "mode" of that key, and each mode has a name:
More reading: Wikipedia - Musical Mode
A scale is any sequence of ascending (or descending) notes that can be used as an "organizing structure" for a piece of music. There are many types of scales, including diatonic (the "standard" in Western music), chromatic (containing every half note in an octave), whole-tone (containing notes a whole step apart), and pentatonic (the pentatonic formed from C major uses the "black" keys on a piano). They can even include quarter-tones and other microtones in the music of other cultures, such as Middle Eastern or gamelan music.
A mode is a very specific type of scale, with origins lying in Ancient Greek music; the names for the modes come from these traditions. Although they may look at times like standard diatonic scales, their origins are very different. For instance, the Lydian mode, which "looks" like C major, is actually F, G, A, B, C, D, E, F rather than C, D, E, F, G, A, B, C. The difference means that C major is whole-whole-half-whole-whole-whole-half, while Lydian mode is whole-whole-whole-half-whole-whole-half. (If transposed to C major, the mode would be C, D, E, F#, G, A, B, C.)
A good example to listen to is Bruckner's Os justi meditabitur, which is written entirely in the Lydian mode—but essentially notated in C. It doesn't "sound" like it's in C, though; that's the result of its modal foundation instead of a standard diatonic scale.
A mode is a type of scale.
Consider a Diatonic scale. For each tone in the scale (1, 2, 3, 4, 5, 6, or 7), you get a different mode when you start on that note.
Think of C Major:
C D E F G A B.
Notice how there are 7 notes.
There are 7 modes in a scale, one for each note of the scale.
Modes are the scale, starting from a different note. In this instance, C Major is the parent scale and all the modes are derived from it.
So using the notes of C Major we get:
C Ionian (The same exact thing as C Major) D Dorian E Phyrgian F Lydian G Mixolydian A Aeolian (The same exact thing as A Minor) B Locrian
In theory, if I play D E F G A B C D and the key of the song is in C Major, then I'm playing in D Dorian. However, actually using modes is a bit more complicated. There's as much theory with them as any other aspect of music. This is just an overview of what they are. In fact, you can spell modes relating them to major scales, which gives you formulas that you can use to make modes on the fly.
Try to stay with me.
So if I have D Dorian in C Major, I can think of it as D Dorian in C Major. But what if I don't know what the key is, or I don't know what the mode is? This is where parallel majors come in handy.
Parallel majors are just the major scale of whatever mode/scale you're playing in. So if I'm in D Minor, D Major is the parallel major. G Lydian's parallel major is G Major.
So let's say I have a lot of notes:
F G A B C D E F
And I don't know the key of the song. I can compare this to the parallel major just to see how it's different, and this will tell me which mode it is because every mode has its own formula, which make it unique. The notes of F Major:
F G A Bb C D E F
So comparing the notes that we had before to F Major, we realize we have a #4 (Bb to B). Lydian is the mode that has a #4, so the first group of notes I had was F Lydian!
You can repeat this process for any mode. In fact, I recommend spelling each mode against its parallel major to understand how they're different yet the same. Here's each formula for the 7 diatonic modes of the major scale:
Ionian: 1 2 3 4 5 6 7 (This is just the major scale)
Dorian: 1 2 b3 4 5 b6 7 (So D Dorian differs from D Major because D Dorian has a flat 3 and 6 while D Major doesn't)
Phyrgian: 1 b2 3 4 5 b6 7
Lydian: 1 2 3 #4 5 6 7
Mixolydian: 1 2 3 4 5 6 b7
Aeolian: 1 2 b3 4 5 b6 b7 (This is just the natural minor scale)
Locrian: 1 b2 3 4 b5 b6 7
While the terms can be used fairly interchangeably, that only speaks to the practical applications; where each comes from is slightly different.
A scale is an ordered sequence of notes with a start and end. A mode is a permutation upon a scale that is repeatable at the octave, such that the start and end points are shifted.
For example, the major scale is repeatable at the octave. Since it contains seven distinct pitch classes within the space of an octave, there are seven possible modes on the major scale. These are known as the church modes, and have different names like Ionian, Aeolian, Dorian, Lydian, etc.
Major pattern in half/whole steps:
2nd mode of major (Dorian) in half/whole steps:
Modes in general, however, are not limited to this. You can extend the definition to other octave-repeatable scales. For example, consider the harmonic minor scale. Its pitches don't line up with those in the major scale or any of its modes, but the scale is repeatable at the octave. You can play in modes of the harmonic minor scale.
Harmonic minor pattern in half/whole/aug (augmented 2nd) steps:
5th mode of harmonic minor in half/whole/aug steps
Obviously, when you take a mode of a scale, the result is still an ordered sequence of notes; so you can call a mode a scale. And, for all scales that are octave-repeatable, you could, if you like, refer to them as modes. But for the reasons stated above, they are -not- the same in how they are generated. It is possible to have a scale that is not octave-repeatable (that takes up more notes than can be fit into an octave), but all modes come from a particular named scale (including the 1st mode of each scale, which is isomorphic to the scale).
Aeismail is confusing - using the black notes on the piano will give either F# maj.pent (Gb maj pent.) or D# min.pent (Eb min pent). However, all the basic modes (dorian, mixolydian etc., ) are still scales - groups of notes that work with each other- but the only modes that are scales as we know them are the Ionian as in major, and the Aeolian as in the natural minor. It's interesting that a composer may state 'this is in D dorian, then writes in the key of Cmaj !! Also don't be messed up with the other naming method - the dorian OF C is in fact, D dorian !!
The seven inversions of the major scale are the modes. So every mode is a scale; but not every scale is a mode (eg. the harmonic minor scale is not a mode).
I think the musical answers already given are the most useful, but the dictionary answers may also be informative. From the Oxford English Dictionary (OED) we get:
Scale: Any of the graduated series of sounds into which the octave is divided, the sounds varying according to the system of graduation adopted.
Mode: A scheme or system specifying the disposition in a scale of the constituent notes of a melody or harmony; spec. each of a conventionally agreed set of such schemes or systems.
The OED also notes each word's first recorded use (with this meaning). For 'scale' it comes in T. Morley's 1597 'Plaine & Easie Introd. Musicke', where he writes "Here is the Scale of Musicke"; while for 'mode' it is C. Simpson's 1667 'Compend. Pract. Musick' where he writes "That which the Grecians called Mode or Mood". The next printed occurrence for 'mode' listed in the OED is A. Malcolm's 1721 'Teat. Musick" where he writes "I would propose the Word Mode, to express the melodious Constitution of the Octave"