The web abounds with questions like "Would a knight beat a samurai?" "Could Batman take out Wolverine?" and our current line of inquiry, "One-handed longsword versus one-handed war axe?" The answers for this last question are no more clear than for the first two.
What's clear is that the longsword is more versatile and more likely to do damage on any one hit than the axe. It has two long striking edges and a point. A backstroke, thrust or a mid-weapon stroke with the axe ends up hitting its victim with a blunt instrument. The same hit with the sword cuts or stabs the opponent.
At the same time, the axe is spoken of favorably against both armor and shield. Its small cutting surface and heavy head puts a lot of force and bite into its stroke, so it was relied on in the late Middle Ages to pierce heavy armor when swords could not. A lot of combat tricks with the axe involve hooking a shield or striking down over it, speaking to its effectiveness in that department. In fact, the axe and similar weapons (mace, pick, lucerne hammer) became popular again in the late Middle Ages because armor had gotten too heavy to pierce with a sword. While these axes were most often two-handed, they were also more versatile than the old Viking axe, being armed with spikes on the top and back of the head for credible thrusting and backstroke damage.
D&D is not Runequest or The Riddle of Steel - a truth that the literature of the latter game in particular seems to take a great deal of pride in. D&D's combat system was scaled down from wargaming, not worked up from years of practice in reenactment combat. D&D is meant to be a game with abstract fighting, where the shortest interpretation of a combat round is a glacially slow six seconds (count it out while imagining someone coming at you with a sword...), cut down from a ridiculous one minute in many early editions.
(I mean, the main game purpose of a combat system broken up into turns is to let someone flee when they find they are outmatched. Would it take you sixty seconds to realize that the fight is too rich for your blood?)
No matter how many moves and feats got tacked on in later additions, individual tricks, strikes, parries, feints and footwork are not really part of the D&D combat system. It is a game with fighting, but the game is not of fighting.
This means that any flavor we put onto the sword and axe to give them a feel of their own can't be on the same level as Riddle of Steel. We are trying to keep the game mechanics-light, so extra dice rolls and tables are not allowed, and hit bonuses and penalties should be kept easy to remember. We are trying to keep the game character-light, so feats and options are not in it either.
One solution would be to give each weapon a bonus or penalty in hitting and in defense (armor class) to balance out differences in damage, reflecting their ease of handling and lethality. Another - probably the best thought-out aspect of AD&D's overly complex weapon rules - is to give the weapon different damage dice against large vs. small/medium creatures. This was a great way to get across the effects of long blades versus concentrated striking areas. Unfortunately, longswords got a ridiculous bonus against Large creatures (1d12 up from 1d8) making them hands-down the best one-handed weapon in the game, balanced only by their cost.
One design concept I'm very fond of is Quiddity. Nothing to do with Harry Potter, "quiddity" is a formation from the latin "quid" (what) and refers to the essence or distinguishing trait of a thing. Bowen Simmons, designer of The Guns of Gettysburg wargame, brings up this concept in explaining his innovative choice of a reinforcement and objective system.
Although there are many things that distinguish swords and axes, the quiddity of the sword is Versatility. The quiddity of the axe in all historical eras is Overcoming Defenses.
To make sense of Versatility, here are a few situational combat rules.
* Characters in close physical contact with their foe may only attack with daggers or bare hands. Closing to contact against an aware, armed opponent requires the attacker to give up his attack and to not be hit by that opponent. Leaving contact requires the same sequence.
* Characters in tight formation (3 to a 10' frontage) or in a tight passage (less than 5') may only use thrusting weapons in melee: spear, dagger or shortsword.
With these rules in mind: In close contact, a longsword or shortsword in hand can be used to cut like a dagger, doing that amount of damage (1d4). In tight quarters, a longsword can be used to stab like a shortsword, doing the shortsword's amount of damage (1d6).
What does the axe get? For every 3 points of opponent's physical armor class, not counting dexterity or magic bonuses, +1 to hit.
Using Swords and Wizardry as a base: +1 against AC6  and +2 against AC 3 . A physical AC0 will almost never happen, when even dragons in that game are only AC2. Judging physical AC will require a little more ruling and common sense than usual; most monster writeups for old school games don't break down AC that way. This is something I started doing in Varlets & Vermin because it's absolutely essential information to have for doing combat rulings on the fly.
These are some subtle differences. They seem to give the axe a very nice advantage at first, until you consider the typical dungeon environment of an adventure. Creatures dropping on you all the time, running up, jumping out of treasure chests, brawling kobolds trying to take you down ... a good referee will provide plenty of opportunities for close contact. Even more so, narrow passages. And when you consider that you can stand with two instead of one spearwielders in a 10' corridor, the longsword seems like the better choice underground. But the axe comes into its own in a room fight, can hack through doors or beams if needed ... In fact, there's a good case for carrying both. Just watch your encumbrance ...
Researcher in social science and appreciator of hexy wargames, role playing games, interactive fiction, board games, and CCGs. I've learned a lot doing design and rules editing work on the side for the Legend of the Five Rings CCG. But this blog is mostly about other things.