Fint, I have done this a few times during end of January and beginning of February. It can take 5 to 6+ hours to drive from SLC to Jackson, Wy. There is a VAN that leaves from the SLC airport everyday around noon that goes to Jackson, WY . If there is a storm Teton Pass may be closed.
Difficult to answer the question is it safe to drive. Lets just say its not as bad as that Discovery show ICE Road Truckers. It can be a challenging drive. I have driven it twice when it was snowing a bit and there was over 6 inches of snow on the roads at times. I know people that live here in upstate NY that have done the round trip drive 5 times in all sorts of weather and may do it again this year. If you don't have a lot of experience driving in snow, then I would say you should not make the drive from SLC to JH.
Here is one plan that we have done multiple times. We fly into SLC, rent a car at the airport, drive less than an hour north to Ogden, Utah and find a hotel for the night. Get up early in the morning drive less than an hour to Powder mountain near Ogden and ski there until about 2am and then drive to Jackson.
Another option that I have done is to fly into SLC and ski there for three days, on the fourth rest day take the VAN from the airport at noon to Jackson, ski at JH, then fly home from Jackson. Because we had the time and we wanted a challenge (plus bragging rights) we took the SLC TRAX and bus from our hotel south of SLC to the airport and it cost us $1.40 with transfer.