1. Strength. Both Yoga and climbing require physical strength and mental perseverance. Seems like an obvious point that climbing requires the ability to hang onto small holds, yet the greatest strength a climber needs is one of the heart, or mind. The fear of falling, failing is inherent. Death from a fall is something born into our DNA to avoid. So too in Yoga, the mental tenacity required to hold after exhale while in a particularly challenging pose can be quite counter to our inherent desires.
2. Steadiness. If you begin to get the Elvis leg shakes in climbing, it is pretty clear you will soon be falling. In Yoga, these shakes are also a reflection of the mind's resistance to the posture. Riding effort and ease are key to success in either pursuit.
3. Discipline. You do not get to climbing at any significant level of difficulty without practice and attention to some degree of training. So too in Yoga. The more challenging postures like handstand, or headstand for a significant amount of time are not mastered without practice.
4. Being in the present. You could also consider this to be focus. In Yoga it is called dharana, or concentration. If at any point the mind wanders from the task at hand either in climbing or in Yoga, this is not close to where you either practicing Yoga (union with your soul or spirit), nor are you going to have success challenging yourself in climbing.
5. Breath. If you are not connected to the flow of your breath as you move through postures, it is not Yoga. When you stop breathing while climbing, you will not be climbing much higher.
I noted these things as a list, but if you have done any climbing or Yoga, you will know that they are feeding on each. No breathing, no steadiness. Limited strength, or fatigue will lead to an inability to be present, to be steady and to have discipline. And so on. There are more commonalities than these five, but I do hope you start thinking about even one of these at a time and see if it helps either your asana or your climbing.