For crying out loud, don't do that thing where you hold the wood up with one hand and chop down with your hatchet while at the same time letting the wood go. That is just asking for problems.
I don't disagree with Codger's axeman ship, but the simple physics of it is that you once you start the swing, the momentum is almost impossible to overcome with an axe. Of course, the safety practice that Codger is talking about is setting up the right conditions before you start your swing. This entails, using a level splitting block so that your piece of wood being split stays upright. It entails using a piece of wood that is capable of standing on its own. It entails the user understanding where the axehead is going to end up during a miss and having the precision of technique to perform a swing to match his mental calculations.
I agree with Codger that if you follow the right technique, using an axe can be safe and effective and it will certainly process wood faster than baton splitting. However, baton splitting can also be effective in situations where it is difficult to establish the safe parts of using an axe. For example, with a baton, having a splitting block is less important and might even be completely circumvented in some cases. Need to have a self standing piece of wood is less important. The baton can be corrected part way through the process and you can use controlled swings to test the wood before attempting to commit to the split. For people who have a bad back, swollen disk etc., swinging an axe is pretty much impossible but they can often kneel, back straight, and baton with little pain.
I agree that you can't process as much wood by batoning and you will more likely process smaller pieces by batoning than you would with an axe. I've also come to the opinion that most people must make tremendously larger fires than I do because of their demands for an axe. I agree with the statement that often times you don't even need to baton the wood. You can crack the wood over your knee or in a tree crook. There are many ways to get that wood for a small personal fire and baton splitting is one of a variety of techniques.