I have not done the route from Cooke City to Cody, so cannot advise on it. We were travelling from Yellowstone to Billings. We decided to cross the Beartooth Pass, since it would take us to over 10,000 ft. above sea level and was marked as a scenic route. It was our first time at that altitude without being in an aircraft.
It took us much longer to get to Billings that day than expected, due to all the stops for photos and to admire the view. There were flowering alpine meadows, mountain peaks all around us, lots of hairpin turns and fantastic vistas, especially from the top. On the downward leg to Red Lodge, I kept praying that my brakes didn't give out!! We did the trip in mid-August in 110F heat!!
It still ranks as one of the most spectacular roads I have ever driven. To put that in perspective, I have driven the full length of the Amalfi Coast highway in Italy, all three of the corniche roads from Monaco to Nice in France, the Cabot Trail in Nova Scotia and the Pacific Coast Highway from a point near Eugene in Oregon to Los Angeles...plus many more too numerous to mention. The Beartooth Pass road ranks right in there with the best of them as both a scenic route and a fun road to drive. I would do it again in a heartbeat, if we are ever back in the area. Highly recommended!!