I heard the term used to mean the same thing, really...it just means a really well trained horse; one you "should" be able to put anyone on and expect that horse to perform reasonably well, regardless of his rider's experience.
For example, my mom's old Arab gelding is what I would consider pretty bombproof...he is good with any age passenger, and is good no matter where you take him.
My mare, while she is well trained, is not bombproof yet. While I can ride her and do tons of things with, I can't put just anyone on her and expect her to act the same way. She would get too confused still, and rather than risk someone getting hurt, if I put someone on her, it's always on a lunge line, where I control her feet, and the rider can concentrate on learning how to balance.
"The ideal horseman has the courage of a lion, the patience of a saint, and the hands of a woman..."