A "sporty family" is a family that enjoys playing and/or watching sports. A "sports family" could be a family whose members are well-known or notable sportsmen or women.
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AnonymousSporty familyA family that is fond of sport or outdoor activities.
AnonymousSports familyIt's not very natural, but I would interpret it as a family that is famous for their participation in professional sports teams such as the Fittipaldis or the Andrettis or the Unsers.