Without knowing your medical history and seeing you, it is impossible to tell you what is wrong with your hand. But, in general terms, sometimes a crushing injury to a hand can cause a long-lasting problem called a neuroma, a bundle of nerves that have developed into a growth. For some people, a neuroma can cause problems such as pain and stiffness, and intolerance to the cold, as you describe.
The best thing to do would be to go to a doctor and have him or her examine your hand. Only then could a diagnosis be made.