The Ring and bar muscle up are both pretty complex movements when done in CrossFit style, with a kipping motion. If you are looking to compete in a CrossFit competition then they are inevitable. If you are not looking to compete I would suggest watching out with performing this exercise as the risk-reward ratio might just be out of balance.
The amount of mobility, stability, and strength needed to execute this exercise effectively while gaining the benefits are vastly underrated. People tend to jump into this movement too fast which oftentimes results in shoulder injuries. After all, it´s not a surprise when people sit all day with their shoulders are rolling forward and at the end of the day, they all of a sudden expect to just move through large ranges of motion without any restrictions.
Ring muscle-ups place a higher demand on shoulder stability in comparison to the bar muscle-up. The rings, of course, move which makes it more difficult for the shoulders/body to complete the movement without the rings moving away. With a bar muscle-up, you know where the bar is and if it is a good bar...it should stay in place.
As talked about above, the rings move while the bar does not. People can take advantage of this by moving the rings in the optimal direction. What you´ll see with the rings is that people pull themselves through the middle instead of with a bar where people need to move: back→ up→ and over. The bar should be at least under the chest during the pull for the muscle-up on the bar to be successfully completed. With the rings, the transition is about mid/low chest hight. So you can see that during ring muscle-ups the pull is lower than the pull of the bar muscle-up.
During the bar muscle-up, the hands are on a bar that does not move, whereas with the rings the shoulders can be in a more dynamic position. This makes it easier for the shoulders in some parts of the muscle up to move. So when the hands are fixated on the bar, the scapula should be working optimally. Even a slight imbalance or weakness can have tremendous effects on the body. Also, the body has to move around the bar/the fixated point whereas, with the rings, individuals can manipulate it´s position to their advantage. The body does not need to move around a specific point.
So...which one is easier?
It depends, if you are good at doing pull-ups and have a strong back but your shoulder stability is shitty then you´ll probably prefer the bar muscle-up. If you are good at moving your body in 2 different directions, then you´ll find the bar easier. If you are bad at doing high pull-ups, the ring muscle-ups will probably be easier for you.
What I often find is that men who have a lot of strength accompanied by bad coordination find the bar muscle-up easier because they can push into a non-moving object. In their opinion, there is less coordination required because they can easily compensate with their strength.