Actually both offer similar procedures.
The procedure they both offer is a glottoplasty. Dr Kim (Yeson) and Dr Haben have different methods though, and it's hard to tell how different the results are, if at all. Dr Kim uses permanent sutures. Not sure about Dr Haben. Basically they cut a membrane off your vocal folds and suture them together. They heal together and the vibrating portion is shorter and the anterior commissure is higher.
Both do this procedure endscopically meaning they go through your mouth and don't make an incision in your neck.
There are other surgeons who do this who may be not as well known such as Prof Marc Remacle in Belgium.
Dr Haben also offers something called a cricothyroid approximation or CTA. You can look that up to see what it is. This one requires an incision in the neck. He also offers a trachea shave to reduce the appearance of your adam's apple.
Honestly the major difference between the two for most people boils down to travel. For Dr Kim you have to travel to South Korea and stay 9-10 days. For Dr Haben you have to travel to upstate NY, but if you are stateside this is easier and cheaper.
But as far as I've seen here, both produce similar results and neither one seems to be "better." I went with Dr Kim because I was inspired by Jenny and J-Mi, plus I just wanted to travel.

I was also impressed by the modern facilities. But if I had to choose again I may have chosen Dr Haben due to travel concerns.
Hope this helps you make a decision.