Gotta love the comments from those who have no experience with either. I'll never understand that, unless it's simply some need to be part of a conversation.
I currently have both, as well as Dan Wessons, Colts, and Remington. In the past, I've also had Springfield. I may be forgetting something. Of the two mentioned by the OP (Kimber and Sig), both are lovely, accurate, and reliable. The Kimbers have had a slight edge on fit and finish over Sig, in my experience. You should feel confident purchasing either one. Oh, a few posters have mentioned Colt. Obviously another fime 1911, and I love my Colts. I prefer Kimber and Sig a tiny bit more. YMMV