There are gazillions styles of yoga, and functional yoga is not one of them! The term simply means that, regardless of the style you are practicing, you are alert for the specific characteristics of your body and feel confident to adapt the poses in order to enjoy their benefits without getting hurt.
Most yoga styles are taught according to the rules of alignment, which assign clear rules for where to place every single part of the body. The poses are defined by those aesthetic references, and everybody should aspire to perform them in the same way.
When practicing functionally, we are less concerned with how the pose should look and more interested in what the pose should be doing for us. We recognize the immense variation in human anatomy, from the shape of our bones to the orientation of our joints, from the constitution of our tissues to our background and temperaments. We understand what is the function of the pose and learn to find it within the body that we have (instead of struggling to achieve a body that may never be ours). It is a profound shift in approach, one that has completely transformed my own practice and life.