The main considerations that come to my mind are money, risk and lifestyle.
Money: Private training is quite expensive ($25K+) and will probably have you in a low-status, low paying jobs for a number of years before making the majors. Military training is absolutely free with decent pay, great benefits and a superb college education thrown in. Your first civillian job will likely be with a major as well.
Risk: The military is a big commitment with no guarantees. If you lose your medical or washout, you're stuck with them for four years. Upside is that if you do make pilot, you have an excellent flying job that you can sit on until you choose to enter the civillian market. The civillian route has much more flexibility, less downside risk, and is more accomodating of medical problems or slow learning. Both routes offer a not insignificant risk of killing you in the process; don't know which is more dangerous, though.
Lifestlye: Military life can be hard. You go to combat survival school and eat kill-it-yourself raw rabbit. You're a soldier, they might send you to South Korea or an aircraft carrier. You also get to see the world and fly exciting airplanes. With civillian training you get to maintain whatever type of existence you like.
I could go on, but I need to learn about antibiotics.
Also, look into some middle route like ROTC.