Why are children and adults so different?
GREAT QUESTIONS!!!
Why delay sexual maturity in the first place?
-I believe it is also related to "the best-used" of resources. Firstly, it is unnecessary and impossible for children to have sex (biologically and evolutionarily), they don't have the physicality to sustain sexual intercourse or the ability to bear a children inside their body. Secondly, it is to focus resources on growth. Children uses lots of energy on growth. Human body does amazing jobs on allocating body resources or material so that we are in the best shape possible. If they don't do sex, they can focus more on growth.  

Why tie sexual maturity to a host of other changes? For example, why aren't babies born with adult hair patterns? (Yes, because hair growth responds to sexual hormones, but the response could be different.)
- This one is a hard one. I believe it is just how biologists categorize human growth. We can study how human bodies develop in each stage, which is I believe it's really important on morphology. 

Why the different body proportions?
-Evolutionary advantages!!! It is believed that baby has a greater head portion than adult. That's why people say children learn better than adults. Human values their brains more than other organs of the body, developing intellectual capacity as early as possible allows them to win in natural selection. 

Why lose play behaviour and learning ability as adults?
If I understand you correctly, you are asking why adults lose their play behaviour and learning ability as they grow up. For learning ability, I believe it is caused by degeneration, just like the other organs of the heart. We have a mortal body and so once the body hits its maximum growth capacity, it will start going downhill unfortunately. For play behaviour, I believe it is a human behavioural changes due to life experience, social expectation and peer interaction. We lose interest of 'games' as we grow up because they are considered to be childish, silly and not useful.