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 magically confers security on a request, however GET implies some side effects that generally prevent it from being secure.
GET URLs show up in browser history and webserver logs. For this reason, they should never be used for things like login forms and credit card numbers.
However, just POSTing that data doesn't make it secure, either. For that you want SSL. Both GET and POST send data in plaintext over the wire when used over HTTP.
There are other good reasons to POST data, too - like the ability to submit unlimited amounts of data, or hide parameters from casual users.
The downside is that users can't bookmark the results of a query sent via POST. For that, you need GET.