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Beginner here, it seems like a scale just defines a particular set of notes that don't have to be played in any order to me. But if you look at A minor pentatonic scale you have A, C, D, E, G, yet with the same notes you get C major pentatonic with C, D, E, G, A. If they are not supposed to be played in order, how would you differentiate the two scales at all? Or does the order matter? And if you play a scale such as A minor above, can you mix in a note outside of that scale, or does that create a new scale from that point? Like if you played A, C, D, Db, E, F, G, is that just A minor with extra notes or would that be considered an entirely new scale?

A scale is a particular set of notes played ascending and/or descending. French word for scale is echelle, which also means ladder. And, as said previously, we scale a ladder. Best to use one rung at a time!

In an examination situation, particular scales are requested, and generally get played up and down, in note order.

By adding ( or taking away) any note/s, the scale is changed, so it becomes called something different. For example, C pentatonic minor, C E♭ F G B♭ becomes C minor blues when F#/G♭ is added.

Taking your example of pentatonics - A C D E G starting from A is A minor pent., while C D E G A is C major pent.This opens a can of worms with modes, which are covered in many places on this site.

So, basically, a pitch set of notes, which often constitute a key, when played in pitch order, is a scale, which can be one octave minimum, or as many octaves as the instrument will allow.

Beginner here, it seems like a scale just defines a particular set of notes that don't have to be played in any order to me.

It depends. As Tim says, in an examination situation, you play the notes in order, as that's the 'study' form of the scale. But when the scale is used in a piece of music, then as you say, it's the set that's more important, and the notes could be in any order.

But if you look at A minor pentatonic scale you have A, C, D, E, G, yet with the same notes you get C major pentatonic with C, D, E, G, A. If they are not supposed to be played in order, how would you differentiate the two scales at all?

At least when talking about Western music, there is also the assumption that the "letter name" of the scale is also the "home note" or tonic of the passage of music using the scale. So you would talk about a set of notes using the C major scale if the home note or tonic was C; with the same set of notes, you could talk about the scale being A minor if the home note was A.

You can also talk about modes of a scale, which is another way to talk about notes from the diatonic scale being used with different home notes.

And if you play a scale such as A minor above, can you mix in a note outside of that scale, or does that create a new scale from that point? Like if you played A, C, D, Db, E, F, G, is that just A minor with extra notes or would that be considered an entirely new scale?

It could be either - it's up to the person analysing or composing the piece to decide what's the most helpful way to see it. Western music theory tends to like to see everything as at least diatonic, with a presumption towards major and minor, so people are used to the idea of using extra notes from outside of the basic scale.

As for how to differentiate between A minor and C major, you may be interested in the answers to When is a piece in A minor versus C major?

But I wanted to offer one clarification in terminology: the difference between a scale and a collection (sometimes called a scale collection). A scale would be the notes of A-minor penatonic in order: A C D E G A. That collection, meanwhile, would be those pitches, but not necessarily in that order, like C A G E D.

This helps clarify that the A-minor and C-major pentatonic scales are different, but their collections are the same.