Does anyone know the functional difference between these two words? Is there any?

If I remember correctly, 体育 is physical activity as in Phys. Ed. class. 体育馆 is a gymnasium. (Note: the gym you go to to work out is called a 健身房.) 运动 is just a physical activity in the more general sense.

If I wanted to ask the simple question, what kind of sport/ activity do you like (to play) how would I ask that. 你喜欢什么体育？ 你喜欢什么运动？？ neither seems right.

What west said. But 运动 is also the best translation for sport. Your second sentence is fine.

I think the second sentence is fine. I might say 你喜欢什么样的运动？

运动 is sport. 体育 is one of the five sectors of education (五育), which are namely 德育、智育、体育、群育、美育 (moral, intellectual, physical, social and aesthetic education). 体育 can also be extended to refer to all the activities involved in physical education.