Unfortunately, it is not possible to match 100% and have the same results. There are different shaders for the materials, different light behavior ect. In the scene that you have send us, this as close as it you could get.
It looks to me like it’s just rendering in mental ray with more intensity. If you layer your normal map over top of it self in photoshop on soft light mode it will increase the intensity of your normal map. Or layer the base normal map color over to lighten.