I think a big difference is that in sports, although you are aware of internal processes, a large amount of your attention is on external things, as it must be so you don't run off a cliff, for example. When meditating you turn your attention inwards, perhaps at first observing bodily sensations, but continuing inward until the body fades away and only mental processes are observed and there is no attention given to the external senses.
But you are often focused on the present moment, not thinking about the future or the past - which is one thing that most mindfulness training seems to teach.
