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I'm aware that A Minor is the relative key to C Major but how does this change the methods for improvising?
Would you use the Pentatonic as you would with C Major or are there other scales which can be used only with C Major that can't be used with A minor and vice-versa?
The Cmaj scale and the Amin (natural minor) scale have the same notes. However this doesn't mean that the notes are used in the same way. The note A for instance sounds and functions completely differently in the two keys. It will sound colorful and perhaps even slightly unstable in Cmaj but will sound solid and foundational in Amin. When improvising, it can be useful to think in both scales from time to time, mixing the Amin scale with the key of Cmaj and vice versa. Even though they contain the same pitches, the melodies you will come up with will be completely different.
For instance, when I'm in Cmaj, I might decide to think and play in these scales to name a few:
Cmaj
Cmaj pentatonic
Gmaj pentatonic
Amin
Amin pentatonic
Dmin pentatonic
Fmaj pentatonic
Emin pentatonic
Although none of the scales after Cmaj introduce ANY new pitches, each of them gives a different sound and color. Depending on style on context some of these may be more appropriate than others, so use your ear to tell you whether it sounds good.
One word of caution. I do not recommend ignoring mastery of the Amin scale just because it contains the same notes as Cmaj, which allows you to get by knowing just the one scale. This will greatly inhibit your ability to think in a certain key and to come up with varied ideas in that key. So, in short, learn all the scales and decide how YOU want to use them.
IANMT ("I Am Not A Music Teacher"), so take my advice with a grain of salt. Maybe a handful of salt, even :-)
A scale is nothing without context. As you point out, C major and A minor are the same notes. The patterns you use to improvise in any of them (I'm speaking guitar-wise here) could be the same. Having said that, "specific" patterns (i.e., "A minor scale", instead of "C major scale") allow you to accentuate better the strong points of the scale. I.e., the root, the third and the seventh. The root note usually falls below the index finger, and it's easier to get to the rest from there.
I think the best way to experiment with it is to improvise over A minor using a C major pattern. For example, you could do it using a plain Am-Dm-Em chord progression. Try to accentuate the chord notes of Am using a major pattern, and you'll see that they don't fit so well as a minor pattern.
My 2 cents, anyway.
Am pentatonic has exactly the same notes as C maj pent.The main difference is that playing in Cmaj., you'll centre on C F and G mainly, whereas in Am you'll centre on A D and E mainly.By centre, I mean when the tune is e.g. on a G chord, the phrase which will work well will probably start with G or at least reference G somewhere significant.
Am full scale is a little thorny - A natural min. has exactly the same notes as Cmaj., so will generally work really well. A harmonic min. uses a G# instead of G nat. thus it may clash with the dominant chord of the tune.If the tune is using E or E7, then a close match will be G#.That said, the G nat. will bring a bluesy feel to that section, so it's not all bad.
A melodic min., however is another kettle of fish.It can feature Gnat. and G#, and Fnat. and F#, so gets tricky to use. Jazz players like it, though.
Perhaps I use the word improvisation differently to others. If I am improvising with the white notes (C Major, A Minor, whatever) Then I am free to play whatever I want, so for me there is no difference. I am not looking back at what has gone before, rather I am looking forward to what is possible.
