I am a local to the area near Burlington and have several friends and relatives in the area. Burlington is a nice place to live. It is the best of country living. If you want to garden, raise crops, raise animals, and have some of the best weather in Wyoming then this is the place. The Big Horn Basin is one of the best places for agriculture in Wyoming. Also this area has probably the least amount of wind anywhere in Wyoming, which is saying a lot. The closer you get to Cody the worse the wind. My picks for living would also be Worland, Tensleep, or any of the small towns on the east side of the Bighorn basin.
The only downside to living in Burlington is shopping. The people that live there travel to Powell, Greybull, or Cody for the groceries and to Billings for the big stuff. Billings is nice to have close by for that reason, not to mention they have no sales tax. Also there is a VA clinic in Powell. If you have any more questions about the area, PM me.