"walking and talking three year old"
This noun phrase is more specific because it lists the traits of a toddler and specifies age.
"Successful entrepreneur John Allan"
This noun phrase describes John Allan as a successful entrepreneur. It gives a better indication of why he would want Poe to be a businessman as well as hints at the source of the Allan family wealth.
"unpolished, shorthand poems"
This shows how Poe was unable to devote all his time to writing poetry since he was supposed to be working and studying business.
"the budding amateur poet"
This noun phrase shows the newness of Poe's career as a poet just starting out.