I expected this too, and I was quite surprised to hear the us government had no idea exactly what Snowden had stolen. Kinda sheds light on the "protections" they have in place, in that they didn't seem to be very good at what they're supposed to do.
But the difference was that in this case, the default policy is collection. If you collect everyone's data, then no special effort has to be taken to snoop on an individual's data (besides accessing it). There might be access logs; there might not be.
If there is a process that has to be followed to initiate data collection on a target, there will certainly be records of that process because they will be required to execute the process. It's at least better than having everyone's data sit in a datacenter somewhere ready to be searched by anyone with the proper security clearance.