We can not really compare Yoga and Pilates.These are two different disciplines.
Both of them are sharing common points such as mind and body do not function separately. Joseph Pilates has designed a unique series of life enhancing physical exercices that help to correct muscular imbalances and improve posture, coordination, balance, strength, flexibility and breathing capacity.
Also Joseph Pilates developed several apparatus such as the Reformer, Cadillac, the Chair, Spine Corrector, Barrel, Ped-o-pull, foot corrector. One of my favorite apparatus is the Chair.
From my personal experience the Pilates Method helped me to improve my Ashtanga Yoga practice. During one workshop with Mark Robberds he said “if you want to practice Ashtanga Yoga you have to become physical”. I was happily surprise to see some of the exercices that Mark was given us in order to enhance and work more efficiently into our practice were inspired from the Pilates Method.
When I was practicing and teaching only Pilates my core was very strong but my upper body was weak. When I started switching my yoga practice toward the Ashtanga Yoga I was not able to do any proper Chaturanga Dandasana as my arms were not strong enough.
For about two years I did not practice Pilates at all but mainly Ashtanga Yoga, then I noticed that my core was getting weak even thought I was practicing Bandhas.
As soon as I started to re integrate an evening practice with only Pilates exercices to my daily routine my all practice drastically changed. I was in a safer zone.
However I do not think that we should be mixing these two disciplines. They are acting as a complement to each others.
Pilates Method is all about the core centering, moving from the pelvis, bringing the spine into the “C” shape, working deeply all transverse muscles which really help to improve all jump back and jump front. Also by activating core muscles all backbends will be done in a better and safer way.
In Yoga sometimes we tend to focus more on the flexibility than on strength. Movement of bones occur only at the joints. Muscles are necessary for movements. Muscles are attached to the bones by tendons. Weak muscles around the joints can contribute to hyper flexibility or hyper mobility.
Therefore a regular practice of Yin Yoga can be dangerous for the body as that particular practice is only about stretching and releasing. You can have a Yin Yoga practice once in a while but not a daily one.
Also Pilates Method is such as Yoga moving from and with the breathing.
Enjoy your practice!