From Wikipedia: "Yoga is the physical, mental, and spiritual practices or disciplines which originated in ancient India with a view to attain a state of permanent peace of mind in order to experience one's true self."
"Contortion (sometimes contortionism) is an unusual form of physical display which involves the dramatic bending and flexing of the human body."
In 2008, before starting pole dancing, I had a few months of devoted yoga time, more specifically Ashtanga/Vinyasa yoga. I chose Ashtanga because at the time it seemed the most "hardcore" one (I tried Bikram too, but I couldn't stand the heat). What was I looking for in yoga? Honestly, I wanted to be more flexible. You see I had this desire already back in 2008, but little did I know about contortion or flexibility training, I thought yoga was the way to go. It lasted for a while, then I got bored. I got bored of sun salutations (why so many?), of poses I wasn't getting a sense of challenge out of it, of its philosophy, I wanted it to be fully physical, mind and body work, without the spirituality and the mantras (ommmmmmm).
Yoga is great don't get me wrong, but wasn't for me. In my experience yoga (Ashtanga) trains the mind-body connection, and it gives a solid foundation of flexibility, but it doesn't take you further, because "the goal of this style is not to learn the more difficult asanas but rather to learn to maintain internal focus throughout the practice" (from Wikipedia). It has flow (vinyasa) and develop core strength (bandhas), challenge your balance and create great alignment, but I think it doesn't focus on a personal and unique path of discovering the body and its capabilities (my interest).
Contortion, first of all is considered an art, and it's spirituality-free. Contortion in my experience focuses on the poses and the stretches that get you to the poses (that make up an act to perform), it's very straight forward. Also, contortion is not for everyone. It's very hard and difficult to learn, especially after a certain age. It can be extreme and scary. Yoga is rarely extreme and scary. Contortion is mainly based on back bending, splits, handstands and such; while yoga I believe it's more about perfect alignment, opening hips, inversions, some shoulder and lower back bending, but there is zero neck lengthening and little middle/upper back work. The breathing is also used differently. Yoga is very popular in Western society, while contortion is rare and not well known. You can learn yoga with an average instructor or by yourself, but you can't learn contortion, because you can get hurt and beyond every pose there is a long preparation for it, which you need an expert teacher to guide you through. I feel so blessed because I met my coach Otgo Waller, who happened to be a Mongolian contortionist (check her page here) and one of the few professionals familiar with working with adults. I owe her so much and look forward to train with her as often as I can. So I'd say yoga's primary goal is not advancing the flexibility but creating a deeper connection with the self through the practice, it's still a good way to get flexible, just with a different approach and a different philosophy. Contortion is pure flexibility training, the deepest way to test your limits and will, and to take down your fears. It also transforms your body completely. To me contortion takes you above and beyond any expectation you might start with, yoga has an ends in itself. But again, this is just my experience :) whatever you choose, make sure it's right for you and it makes you happy.
Me doing yoga
Me doing contortion