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).