I do Pilates twice a week MorComm and I can tell you that Pilates is precise 'neuron to the fiber'. That is brain to the muscle. The Pilates breathing and the control of the core and pelvic muscles is different to the Yoga breathing. Nevertheless, in both, Pilates and Yoga one walks away having had 1 hour of brain oxygenation which results in a harmonious feeling of peace and balance within and without. I work with different types of Pilates equipment, like a reformer and the caddy, as well as other pieces of Pilates machines. I do not like the mat work of Pilates. Pilates uses a lot of resistance work with springs which my muscles so enjoy. In Pilates one learns to control the smallest of muscles, hence the word they use 'precision in movement'. This can only be done from the mind to the body, hence the expression 'neuron to the fiber'. And it really transforms the body alignment and makes one 'mindful' of the muscle-skeletal connections and how we move them.
I also have done different types of Yoga for years and have enjoyed them immensely. Especially I like Iyengar yoga, it is slow and and I felt that I had more control than doing Ashtanga Yoga which moves fast, hence called 'power yoga'. Yoga is more 'meditative' and uses the Sunskrit exotic language. In addition, Yoga has a strong spiritual emphasis too, it encompasses meditation and even can have a religious emphasis. So the difference between Yoga and Pilates is that Pilates works very well to balance, body and mind without the strong spiritual emphasis. It is more like a bi-dimensional approach to physical activity.Yoga, which means 'union', uses a more tri-dimensional approach of body, mind and spirit.