There's definitely a mind-body connection and a very similar fluidity in both. Pilates offers a line of equipment that doesn't exist in yoga, so it provides a different angle: You're doing exercises with the assistance and resistance of springs and pulleys. The springs may assist you or they may make an exercise more difficult, depending on the exercise.
Those who take the time to study the ancient art of Hatha yoga understand that there are some specific benefits of becoming adept at practicing this yoga style. A main benefit stems from one of the pillars of the Hatha style: breathing. Controlling breath can help improve oxygenation of the body and alleviate stress in some forms. Hatha yoga will help you experience freedom from preconceptions about yourself allowing for the possibilities of growth, acceptance and change. A program of yoga poses, breathing and relaxation significantly reduces joint tenderness and swelling for people with rheumatoid arthritis (RA), osteoarthritis, and fibromyalgia. In addition, it has been shown to be helpful in reducing high blood pressure. Other elements of Hatha yoga help with relaxing the body and fighting modern world stresses. As a preparation for intensive meditation, Hatha yoga has a lot to offer a modern Western audience.
The difference between Pilates and Yoga lies basically in the approach behind the exercise. Pilates encourages body awareness and conscious movement, aiming at grace in every step. Pilates is very popular with dancers as a form of exercise. Yoga has greater emphasis on the development of a spiritual attitude. Yoga exercises lengthen the spine to release spiritual energy and Pilates concentrates on spine lengthening, body balance and fluid movement.