Corporate Captain reported DIJ Runway 04/22 was recently repositioned and when there is snow; the runway markings are obscured and it becomes hard to distinguish between the old and new runway locations.

Narrative

I was flying into Driggs; ID. The weather was lower than reported. We initially flew the GPS-A approach; with the intention of circling to land on the preferred Runway 22. We descended to the minimums for the approach and only saw the airport when we were right overhead; and definitely did not give the flight visibility to circle. I executed a missed approach and then requested the RNAV 4; with a light tailwind. The weather appeared to be right at minimums. I acquired the snow covered runway at about 400 feet and 1 mile. When transitioning to visual flight I utilized the PAPI and the aircraft Head-up Display (HUD) to pick an aiming point on the runway. Driggs has recently repositioned the runway; shifting it approximately 2000 feet to the northeast.When there is snow cover the space where there used to be runway is almost indistinguishable from the actual runway. The runway markings were obscured by snow. It would have been very easy to have landed short of the paved runway surface. Especially when landing with a tailwind; and trying to maximize available runway. I was aware of the runway shift project; so was able to process what I was seeing in a very short amount of time; but I think that a pilot with less experience may not have been able to process the visual picture as quickly.My recommendation would be a note on the airport diagram to indicate that the runway has been shifted; and that in times of snow cover; the area short of the runway could be incorrectly identified as a landing surface.