In the 15th century, a disciple of Swami Gorakhnath wrote a book called "Hatha Yoga Pradipika", which in effect created a new system. This system was very strongly influenced by the more esoteric practices of the Siddhars (the pradipika begins by listing the great siddhars, starting with Siva, whose philosophy it says it follows). Hatha Yoga had practically died out by the 20th century, when one Tirumalai Krishnamachary revived it. His hatha yoga is largely based on the theories of the Pradipika. However, it greatly expands the rather limited collection of asanas which the Pradipika describes (no more than 15 or 16).
There is yet another type of Yoga, called Kriya Yoga which was developed by Swami Yogananda, based on the teachings of Shyamcharan Lahiri.
It is these latter two systems that have spread themselves to the west (and in India), and all too often by people whose primary concern is feathering their nests. Bhakti is not as important a part of these systems as it is of the Gita's concept of yoga, so it is quite easy to cut it out and prepare a more easily marketable product.