You should probably clarify what you mean by "favor".
I'd be careful about saying Vikings "favored" axes over swords. There were definitely very well known (and expensive) presumably scandinavian swordsmiths, and a well crafted sword was an item of prestige above an axe. Axes were certainly more common on the battlefield in Europe post-migration period than they had been prior. It's much easier to make an effective axe than a sword in terms of materials and time.
There were many different cultures that used axes extensively over a long period of time, if that helps answer your question. The first that comes to mind would be the Scythians and other groups from central Europe/northern Persia during their heyday. The scythian battle axe (called a sagaris) had a very small head compared to what you might expect though. Their warriors were often buried with weapons/armor, and axes are predominant along with arrows. There are also some shown in greek artwork from the period like this.
The axe never really died out on the medieval battlefield, but it would evolve. On horseback, the horseman's axe was used well into the 15th century, and on foot halberds and poleaxes would come to be very effective anti-armor weapons. They may not have had the romantic imagery that swords inspire, but they were certainly effective, and soldiers armed with polearms would be a staple of many armies all the way through the renaissance.
You're not comparing like with like, at least not always, in this question.
Sword is a sidearm primarily (well, until we get to greatsword territory), you usually don't go into battle hoping to use it any more than a modern soldier hopes to use his pistol. Primary weapons of the time were either projectile weapons or polearms.
Even vikings themselves used spears as their primary weapons, with swords being drawn only when the shield wall fell apart and skirmishing/slaughter of routing people started.
Only exception to this was occasional use of swords and shields in first rank, with spearmen being in the second, the logic probably being that front line doesn't need to attack that much, just hold, and swords parry a lot better than spears when this close to enemy. Picture: http://manuscriptminiatures.com/4661/12562/
As for axes, there are two types - pole axes and one handed axes. poleaxes shouldn't really be compared with swords, but with spears and other pole weapons, and they remained popular well into the middle ages: http://manuscriptminiatures.com/4287/9662/
Then there are one handed axes. Their chief problem is that they really, really suck when compared to the sword if you don't have a shield - they are shorter and have most of their weight at the head, so they are pretty bad at parries, they don't have any hand protection at all, and can be grabbed by your opponent much easier than a sword can.
Once you have a shield to use for defense, the situation gets a lot better, good enough that sword or axe is a question of personal preference - axes can do things swords really can't, like hooking. That's why many people who used axes, sometimes even poleaxes:
used shields:
had some pretty heavy armor: http://manuscriptminiatures.com/4673/8014/
or both: http://manuscriptminiatures.com/3976/10971/ http://manuscriptminiatures.com/4648/12021/ http://manuscriptminiatures.com/4135/7289/
or were too poor to afford a sword, or simply grabbed what was available nearby, in a pinch, anything goes: http://manuscriptminiatures.com/4724/12504/
In some sense, axes were more used than swords because they had some examples in more weapon categories than swords - primary weapon of a dismounted knight circa Agincourt was the pollaxe, after all.
One final picture with a crowd scene where you can see solders equipped with both axes and spears in about 50/50 split: http://manuscriptminiatures.com/4203/9938/
