I am a bit contrary on this. FMJ was used by the military since ? And we can kill people but not animals with it?
What 44AMP said. The military uses FMJ because international agreements decided that it was more humane for use on humans because people are less likely to be outright killed or disfigured when using FMJ's, and thus keep the body count lower than it would be otherwise. In other words, they're used precisely because they are ineffective. There's also a theory floating around that states that a wounded soldier is better than a dead soldier, because they tie up medics and possibly other soldiers who would otherwise be shooting back at you.
On the other hand, when we're shooting game, we want a quick, humane kill because the kill itself is the objective. We don't want the game to suffer, because we aren't going to fix it up and send it back home, it's destined for the grill. So something that maximizes quick lethality is far more appropriate.