Steve,
I don't know much about BJJ, but I thought the gracie brothers made some interesting comments on this subject:
There are so many things that are taken for granted in a sport. You're fighting for points. You're fighting in a ring. You'll only be fighting one opponent. You'll be fighting on a mat. You and your opponent will be (or will not be) wearing certain clothes. You won't be wearing shoes. Strikers will be wearing gloves. There will be rules about how and where you can attack; usually you can't attack targets such as the neck, groin, eyes, or spine - often, you cannot kick a downed opponent, or you may not even be able to strike at all. You will be fighting opponents of a certain nature, of a certain culture. If you're in an MMA context, your opponents are going to be concerned with one thing; fighting and winning. They will be popular, professional fighters. They
usually will be big, buff guys who wax their chests, arms, and legs, wear tattoos, fight in their underwear, and train simple and effective arts not for the sake of the art, but simply as a quick and effective method of learning to win the game against other very similar opponents with the same goals and methods. And if you want to be popular and successful in that particular sport, you have to fit the culture and fit the game.
Generally, these games are set-up to prolong combat and make it interesting and entertaining to watch, and fun to participate in. The participants aren't trying to kill or seriously injure one another. They will start square off, in a very controlled environment, proceed with care and feel eachother out, and engage in a physical duel. This is in contrast to a serious fight, in which two opponents close very quickly and very violently. A brief, violent and vigorous exchange of blows, and one party or the other goes down. There is no feeling out your opponent. There are no rules or equipment to prolong the fight and make it safe in any way, and there is nothing that can be taken for granted; weapons, multiple opponents, shoes, bare knuckles, low kicks, pavement and concrete. The very, very specific set of circumstances that you train for in competition become irrelevant, and in many cases, if you think like you do in a competition, you will be in trouble.
Or so the story goes. But like most martial artists, I have no actual experience with actual fighting, so it's all just theory! However, I choose to put stock in the views and accounts of certain practitioners who had their arts tested in self-defense and skill comparison fights of the mid 20th century.