OK, I responded a couple years ago, and some things have changed personally, so here's an update....
It's difficult to draw a clean comparison, because the AASR consists of continuous degrees, culminating with the 33rd. The York Rite in the U.S.A. on the other hand, consists of 3 distinct bodies: Royal Arch Chapter, Council of Royal and Select Masters, and the Commandery. All bodies are independent of each other, are co-equal in stature, and yet some build on others. You have to be a member of the Royal Arch to join the Council and the Commandery, but you don't have to be a member of the Council to join a Commandery. I've seen in some states where the Commandery has been elevated to a prestigious level above the other bodies, but sadly arrogance and ego tends to drive it. In South Carolina, we seek for unity and equality, viewing all bodies as necessary to gain all of the Masonic Light that the YR has to offer.
In addition to the three bodies are many Appendent bodies that confer additional degrees and honors. Some simply follow on, and others are invitation only organizations. Some delve into the esoteric, some fill in gaps left in the YR degrees, and some are for honor and recognition for service and support.
The KYCH is an honor, but to me it's more of a milestone. You earn it by completing service as Master of a Blue Lodge, High Priest of a Royal Arch Chapter, Illustrious Master of a Council, and Commander of a Commandery. There are specific administrative requirements that can vary from jurisdiction to jurisdiction. And when you serve as a grand head of one or more of the four bodies, you can receive the KYGCH, or Knight York Grand Cross of Honour. For each grand head position you serve, you receive a "quadrant" on your jewel of the color of the body. So when I served as Most Illustrious Grand Master of the Grand Council of Royal and Select Masters of South Carolina, I received the KYGCH with one purple quadrant for having served the Grand Council.
The Red Cross of Constantine is an honorary degree and organization. It is limited in members, and is by invitation only. Like the 33rd degree and other honor organizations, you do not ask for it. I received the RCC last year, and I was told that it is accepted by the Supreme Council AASR as equivalent to the 33rd degree. There are no cross-visitation rights, it's only the "culmination" of standard York Rite Masonry. And like most honor and appendent bodies, there are officer and rankings at the local, Grand, and General Grand levels.
Each of the YR bodies also has "high honor" awards and honors specific to the body. For example, the General Grand Council of Cryptic Masons International has the Order of the Secret Vault that has very limited membership. I was honored this weekend to receive it at South Carolina's Grand York Rite assembly. I did not ask for it, nor was it expected.
The point is that there is no direct correlation between the degrees and honors of the ASSR and the YR, but each does recognize exceptional service and devotion. And remember, that in Freemasonry, there is NO higher degree than the 3rd Degree, that degree on which all Master Masons are on the level.