The brain culls more connections than it makes during childhood and adolescence. But long range axons do increase in calibre, which takes up space and accounts for some of the growth of the brain. It probably accounts for some of the behavioural development as well because bigger axons carry signals long distances more effectively (even more so when they're myelinated).
Very little is known about the functional development, as it relates to the physical development, but we know quite a bit about both behavioural development AND physical development of the brain. Both of them you can, you know, watch.
Storing memories is adequately explained by reconfiguring existing connections. That doesn't mean that's all that's going on, but there doesn't have to be something else happening. The poster I replied to said that obviously the brain is growing new neurons because it develops after birth and we're able to learn. Neither of those things need to involve new neurons.