Key word being ''battle'' axes as I know someone will use vikings as an example which is not quite the case you see aside from the fact that their most common weapon was a spear most of the axe heads archeologists found were wood cutting ones altough indeed they developed lighter, dedicated battle axes later on it was far from some desired warrior weapon (that was a sword). Were there any people or military units who prefered axes over swords? What advantages does an axe have over mace and sword?
First off, you're very wrong about the vikings, bearded axe was used by them and was not a woodchopping axe, let alone tree-felling axe (and those are different types). It simply has unnecessary things on it if you want to use it as a tool, namely the beard.
Next up, there's no such thing as a battle axe, at least not as a weapon type. There are axes designed for battle, yes, but lumping them all into one category will not help anything, since they include everything from one handed horseman's axes, through daneaxes, up to pollaxes, all of which have very different context and techniques of use.
I don't know if most of the axes found were tool axes, because I'm not aware of any archaeology statistics that broad, but the reason behind that is simple - there were more people who needed to chop wood around than soldiers.
Military units didn't exist for much of the middle ages in the modern sense, closest to that were mercenary companies or soldiers serving one noble, but those were all over the place, one knight could have three archers, five spearmen and a crossbowman in his retinue. All of these were equipped privately (either bought weapons themselves, or their lord equipped them), so what weapons they had varied considerably - an archer obviously needs a bow, but whether he has a dussack, messer or a sword as a sidearm was up to him.
As for people preferring axes, well, there were people who used them in battle, and as I said, equipment was up to the individual, so they certainly preferred them over one alternative or the other (some axes were one handed sidearms, some were two handed polearms). To get an idea, here are over 200 illuminations that shw people with axes, spanning 1100 to 1400: http://manuscriptminiatures.com/search/?tags=%22axe%22
Advantages and disadvantages depend heavily on what axe we're talking about, a pollaxe needs to be compared to halberds and spears, horseman's axe to swords, and so on. In general, axes are slower, better at dealing with unarmored or lightly armored (no more than padded armor) soldiers and bad at defense (how much this last part matters depend on what you use for your primary defensive means). Maces are pretty similar to axes, but are geared towards dealing with heavily armored opponents and you don't have to watch your blade alignment when using them.