Verbs can be quite complex in Chinese, because the way you describe the action depends on what you want to emphasize. There are different forms of martial arts where 打 OR 踢 are applicable depending on the style/form and the specific action used. For example, Judo doesn't seem to involve striking an opponent with your foot, and Tae Kwon Do seems to be exclusively using the feet.
In the example you provided, the word 打 in 打篮球 is actually a generic term used to talk about 'playing' or 'participating' in a sport (it can also mean to strike, which makes it a little bit confusing). So you can say play basketball, tennis and even Tai Chi (打太极拳) and it would be perfectly acceptable.
However, if you want to be more specific about the action, 打篮球 and 投篮球 mean very different things because the latter refers to the actual action of shooting a basketball, and you can also use the term 射球 in the context of a basketball game to refer to a shot or shooting the ball.
I think because there is a difference in sports involving the hands, which can be a direct strike or with an implement like a bat or racquet, or using different actions of the hand/arm, it means that there are more ways to describe the action compared to sports where you use the foot or leg (which is generally contacting the ball or person directly), and that's where 踢 is used.