My husband, Tom, has been given both. (I know. Lucky fella!)
I asked him this morning which was better for reading. He said that he uses them interchangeably for nighttime reading. (Which is really the only time he reads for himself and not for work.)
He said that his recommendation, for the price, would be that unless someone really needed an app (needed, not wanted) that the Kindle Fire would fill the bill just fine.
Just passing that along for any interested parties.
I should have mentioned that I have the Kindle. I have found that I really prefer reading actual books although my Kindle fills the bill when traveling or trying samples from Amazon where it is easy to accumulate them in a folder to see if I want to put the library to the trouble of requesting them. Also, as Deacon Greg points out, you can read it in sunlight from the patio where you are lounging in the South of France. Or even in good, ol’ Dallas, Texas.