I have been to both while I was stationed at Richardson.  Alaska is small town everywhere you go.  Anchorage is the largest city in the state but is a small city.  I preferred Anchorage because it had the amenities that were important to my wife and I.  The temperatures are not as extreme as they are in Fairbanks.  Fairbanks can go to as low as -40 and as high as upper 90s.  Anchorage gets as cold as -25 and as hot as 70.  The mosquitos are bad anywhere you go.  The best thing about Anchorage is that the city is set up to be a jumping off point for anywhere in the state.  If you want to hike in the mountains you are very close.  If you want to go on a fishing charter you are about a 2 hour drive, depending on where you want to go out of.  The camping is great and the skiing is great as well since you are only about 45 minutes from Girdwood.  Alaska was the best duty station I have ever had.  As with anywhere though, it is was you make of it.  So no matter where you go in AK just get out and explore.