These two approaches to meditation come from different traditions, are practised 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.