I'd say that if you are a younger person, who is a good player and desires a tougher challenge on a regular basis, and you are not put off by cart golf, your better choice is Bulls Bay. It is a big brawny course that will present more day in and day out challenges to hone a good golf game. And, my guess is the membership and atmosphere is a younger and more active crowd.
If you are a person that likes the fun of a great walk with interesting and clever golf shot making, Yeaman's seems to me to be the better choice. It is dripping with atmosphere of a bygone era, and is just a grand old golf course. I have no idea what the membership social life is like. As mentioned the CCofC may be a notch less golf design quality. Although, since I've last played there, what looks like a very substantial and excellent reno-resto took place and greens and bunker array have really been shined up to look very cool.
I remember playing CCofC once with a past Pres of the club who was a very gracious host. He had to leave after 9 and I finished late afternoon in the early spring, playing the last 9 alone. It was a beautiful evening, and I remember finishing 18 and lingering for a while to watch a group of about 15-20 25-35year olds who apparently came to the club after work, and they all seemed quite familiar and friendly with each other and were in a very comfortable and what appeared usual game of putting for quarters on the putting green. They all sipped their beverages, and in a graceful manner embraced a great evening. I thought to myself that the vibe of that place seemed greatly alluring.