I too was in Barrow in August. It did not snow that day but it did come down to freezing about 4PM.
The place is fascinating for many reasons, not the least of which concerns the native Inupiat population, which is mostly quite
well-to-do because of the oil reparation money that is paid to them.
A tour is a must, but do not be surprised if the Inupiat guide clearly shows contempt for the tourists. In one man's opinion, these are really not very nice people.
AAaLot has it pretty much right with what you definitely want to do, but I would add keeping your eye out the window of the plane coming in to spot the Arctic ice pack, which seems only a few miles off shore.
You should visit Pepe's Mexican restaurant to get the flavor of the place, but leave it at that, the food there is something you definitely do not want to eat.