Someone asked the difference between sport and dress shirts the other day, to which I replied, "Sport shirts have two pockets." Now obviously sport shirts often have more pattern and the cloth is usually not as fine as that used for a dress shirt, but to my mind the principal difference between the two is that a sports shirt has some additional storage as it is often worn without a jacket. A man does need a place for his sunglasses and readers after all, and two pockets give a more balanced look than one.
Now some sport shirts have but one pocket and I think of that as yet another attempt by the makers to save a penny by reducing the cloth required for a shirt and re-using the dress shirt pattern they already have (rather like those two button dinner jackets with flapped pockets and notched lapels). The man who thinks about his clothes will have two pockets for sport and may otherwise have them the same as his button cuff dress shirts since the collar will flap appropriately without stays. That sameness means they can do double duty under an odd jacket as the occasion and temperature warrants.
Sport shirts may be less refined than a dress shirt but that is perfectly all right for many occasions. Linen, oxford cloth and chambray, for example, work well in either guise.
In the photo, Luca Cordero di Montezemolo, Chairman of Ferarri, wears a sport shirt at the Italian Grand Prix.