Tho I didn't play the alpha, just the beta I seen enough alpha footage to give a few examples
- story mode was more complete, alpha players started as Lvl 3 and could only do the investigate the cosmodrome mission, beyond that they could also do explore mode and the strike.
-PvP was all but the same, same maps same one mode. No iron banner event during the alpha tho
So far as I know there was no other things that was different.
English
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Four maps available for the crucible in the beta vs two for the alpha. Plus we could do a mission on the moon that we couldn't in the alpha. Also in the beta you could continue leveling your skills after you reached the beta cap in the alpha you couldn't