Yoga is designed to facilitate serious contemplation. In other words, it tries to provide an integrated approach towards objective and transcendental knowledge. Stronger muscles, stress relief and improved circulation are essentially by-products of practicing Yoga.
Pilates on the other hand, primarily tries to strengthen the core stability muscles of our body to improve the way we function in our daily lives.
Yoga exercises themselves are typically static poses that are held for a period of time, say for one minute each for example. It is the holding of these poses and the transition from one pose to the next that provides stimulation to our muscles.
Because it involves poses that challenge the whole body, Yoga tends to exercise almost every muscle in the body while Pilates simply tries to exercise the deep core stabilizer muscles around our stomach and hips.