These two approaches to meditation come from different traditions, are practiced in a different manner, have different effects on the brain, and are different in the way they're learned.
Transcendental Meditation comes from the Vedic tradition and was brought to the West by Maharishi Mahesh Yogi. Mindfulness comes from the Buddhist tradition, and was popularized in the West by Jon Kabat-Zinn.
Transcendental Meditation is described as a simple, natural, effortless meditation technique that doesn't involve contemplation or concentration. One uses a mantra as a vehicle to let the mind settle down naturally and, ultimately, to transcend thought.
Mindfulness meditation involves training one's mind to be in the present moment. It typically involves passive attention to one's breathing, sensations, and thoughts during meditation, sometimes referred to generically as open monitoring.
In this regard, the main difference between the two is that the goal of mindfulness meditation is to have one's thoughts be on the present moment, whereas with Transcendental Meditation, the process involves transcending thought itself and experiencing a state "pure awareness," in which one is aware but without an object of thought.