Actually, yoga is more studied than simple stretching. It's slower, more deliberate and conscious. It does not have the same random exuberance as an early morning yawn and stretch, for the most part. Sometimes you will press firmly as you stretch, other times you'll stretch with less intensity and be soft. Your intent will change and flow according to the need.
Yoga has many moods.
One is not better than another. The idea is to be increasingly sensitive, appropriate in the moment, so that each moment of practice feels perfect, alluring, desirable - and then to be as wholehearted as possible. The more yoga you do, the easier this will be, and the better you'll get at doing it. The ability to immerse yourself in your conscious experience of the poses and meditations, to be more and more fully present in the Now, is what will cause this awareness to infiltrate naturally into the rest of your life. This, of course, is what it's all about.