We are going to take a flight to one of these two places and stay one night but we are having difficulty deciding which. Right now we are leaning toward Barrow. Is there anyone that can offer an opinion on this and why that is the recommendation? Thanks! If it matters, which I doubt, this will be in August.

Depends on what you're looking for:

Nome -- old gold rush town
Barrow -- native center

I've only been to Barrow. Worthwhile- yes.

I've only been to Nome but I think Repete's point is correct. It really depends on what you're looking for. There is little or no tribal culture in Nome, but lots of history, nonetheless. I understand it's also a top destination for bird watchers.

If your primary motivation is to say you've been inside the Artic Circle, be aware that Nome is not--it's a bit too far south.

You might also check out Alaska Airlines' tours to Nome and Kotzebue; various itineraries but you can overnight at one or the other, visit the other town en route to/from Anchorage. Kotzebue is an eskimo town that is above the arctic circle, so if your aim is to see a native community, stick a pinky in the Arctic Ocean (bit of a struggle at Barrow) and get a real sense of gold rush history, then the Nome/Kotzebue tour would be a twofer.