In both religious traditions, there is a strong emphasis on meditation and the effects of meditative practices to discipline both mind and body. The notion of internal prayer and being devout in one's prayers helps to feed the idea that the body is something that be controlled by the mind. Internal prayer to this end is one means that are employed in helping to fully gain power over this end. There is not much of the idea that the body is something that stands against the mind in either religious tradition. Yet, the belief that prayer and meditation can help to control the ends of the body and, in effect, provide structure to it is something that we see espoused in both religious expressions.
Buddhism is a religion that stresses the achievement of bliss through self knowledge and acceptance. Both Buddhism and Hinduism eschew hedonism and the excessive consumption of the West. There is a strong belief in the body-mind connection: that physical habits strongly influence mental well being. Other methods used to discipline the body include conscious breathing, meditation, fasting, avoidance of alcohol and other toxins, and rejecting physical comfort (e.g. soft mattresses). Rising above physical discomfort is seen as a way to strengthen the individual's will power. One tenet of Buddhism is that the individual must accept that life is suffering and hardship, and this acceptance is seen as a way to move towards enlightenment.
What is popularly called yoga is, in reality, one of the many different practices or paths used for achieving salvation, each of which is called known by different kind of yoga. Some of the main main types of yoga, which are described in Geeta include:
Karma yoga - the path of preforming your duty without being swayed by considerations of personal gain or loss.
Gyan yoga - Learning about the true nature of the universe.
Bhakti yoga - Devotion to some incarnation of god in physical form.
Sanyas yoga - The path of renunciation, or giving up the ownership of all possessions and living a simple life.
It may be worthwhile to note, that though these different types of yoga represent different approaches, they are not mutually exclusive. A person can practice more than one type of yogas, simultaneously and without facing any contradiction.
The popular yoga we know as yoga, which is primarily the path of improving the capabilities of the body and the mind, by exercises consisting of different postures, breathing exercises, and meditation, is a useful supplement to all the other types of yoga.