This has been asked of me -- usually rhetorically -- by students over the years more than once. And of course, I agree... all yoga should be practiced mindfully. Sadly, given today's commodified and mainstreamed practice of yoga as a "workout" or exercise regimen, we cannot simply assume that mindfulness is present in all venues.
But that aside, there is still a difference between yoga practiced mindfully and Mindfulness Yoga. Especially when what is being described as yoga is actually asana practice.
There's an old story about two zen students on retreat wondering if the roshi will allow them to smoke while on retreat. They decide to separately ask the roshi during dokusan. A bit later, one of the students notices the other one sitting in the garden smoking a cigarette. The student goes up to him and whispers, "Why are you smoking? The roshi told me I couldn't smoke." The smoking student asks, "What did you ask the roshi?" The other student responds, "I asked if I could meditate while smoking." "Well," the smoking student answers, "I asked if I could smoke while meditating."
Hopefully, anyone engaging in asana practice, if they understand the purpose of yoga, is doing it mindfully. That is to say, as they move into trikonasana, for instance, they are doing so mindfully, paying attention to what and how they are doing the posture. But as you can see, to say one is doing the asana mindfully is putting emphasis on the asana; mindfully is an adjective describing how one is engaged with the posture.
With Mindfulness Yoga, as the proper noun evidences, the emphasis is on the practice and cultivation of mindfulness through the vehicle of the posture. In mindful yoga, one is practicing asana mindfully; in Mindfulness Yoga one is practicing mindfulness in the posture.
Further, for me, what I coined Mindfulness Yoga is a form or approach to asana practice (including pranayama) based upon the buddha's instructions given in the satipatthana-sutta and the anapanasati-sutta. In Mindfulness Yoga, whether we are practicing slow-movement, restorative, yin or vinyasa-flow, the foundation for practice is the practice of sati, which is the word translated as "mindfulness."
And to be absolutely clear, with the mainstreaming of "mindfulness," much of the practice of sati has been left behind: mindfulness is not the same as "bare attention." That is to say, it is not merely "paying attention in a particular way: on purpose, in the present moment, and nonjudgmentally," as it is most often described by many contemporary mindfulness teachers.
Yes, mindfulness includes and is based upon such an orientation to the present moment: we observe what is happening purposefully, and without judgment or reactivity. But that is only to enable us to see clearly -- to have 'clear comprehension' (Sampajanna) of the present moment. If we judge or react, we are no longer seeing clearly, but rather through the filter of our reactivity.
But, we do this in order to then seen clearly the causes and conditions for the present moment in order to discern what and how we should respond. If the present moment is positive and wholesome, (which is a judgement, after all) then we are determined to continue nurturing the causes and conditions that led to the present moment. However, if the present moment sucks, if is is filled with duhkha and unwholesomeness, then we respond in such a way to cut off the causes and conditions and to not repeat them in the future.
Thus, the full practice of mindfulness is not atomistically looking at the present moment, but is doing so while remembering (which is the actual root of the word sati) what led to its arising. The practice of sati is relational, requiring memory and discernment as we move from conditioned reactivity to creative response. If one is living only in the present moment, there is no relationship possible. Relationship requires memory. Ask anyone who has a family member suffering from Alzheimer's if you doubt that.
So, with Mindfulness Practice as I teach it or in my upcoming retreat at Kripalu, we follow the various practices outlined in the two suttas I referred to above which includes observing and investigating breathing, the positions and activities of the body, the various parts and elements of the body, and the ultimate state of the body; whether the feelings arising are pleasant, unpleasant or neutral and whether arising physiologically or psychologically; all mental activity; and the fundamental categories of the buddhadharma.