Choosing common time or alla breve
From a technical standpoint, there is no real difference between common time and alla breve at half the frequency per beat (e.g. quarter note = 160 under 4/4 or half note = 80 under 2/2). Generally, I found that alla breve tends to be used for relatively fast pieces, especially when individual quarter notes are moving so fast that it becomes unreasonable to count each one of them, but what ultimately motivates a composer to choose one or the other?
Edit: Removed the bit about the Kreutzer sonata, as I realized I had a bad copy and it is, in fact, in 2/2 (as I would have expected).
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I suspect there are as many answers to that question as there are composers. But one of the things that comes to mind is simply appearance. Sometimes alla breve just conveys a sense of faster music.
A good illustration is the coda in the last movement of Beethoven's Waldstein sonata. Have a look at around 21:38 in this performance, where he changes from common time to alla breve and marks the tempo Prestissimo. Of course, he could have kept it in common time, too, and just said to play it faster, but there's a dramatic change in tempo here, and I think it's better conveyed by changing the time signature than simply writing Prestissimo.
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