When comparing an HTTP GET to an HTTP POST, what are the differences from a security perspective? Is one of the choices inherently more secure than the other? If so, why?
Neither one of GET and POST is inherently "more secure" than the other, just like neither one of fax and phone is "more secure" than the other. The various HTTP methods are provided so that you can choose the one which is most appropiate for the problem you're trying to solve. GET is more appropiate for idempotent queries while POST is more appropiate for "action" queries, but you can shoot yourself in the foot just as easily with any of them if you don't understand the security architecture for the application you're maintaining.
It's probably best if you read Chapter 9: Method Definitions of the HTTP/1.1 RFC to get an overall idea of what GET and POST were originally envisioned ot mean.