The value of a sword is inversely proportional to your strength score and alchemy skill. The other perk of swords is that you start with an actual sword with +5% THC and some hardness on top for denting. Since it does flat damage it also benefits more from spamming fast attacks, which conveniently are most likely to hit. On the other hand they fall behind on slower hits.
Axes do more damage than swords as long as your axe does at least 16 damage on the swing, since the axe does 4 damage total. Early on swords are better, but later on you build a fast steel axe and start slapping whetstones on it and it will do 10-13 damage pre-strength, so with 10 str and 6 crafting you will have a 14-18 damage swing, which means you will do +4 damage on average, non-berserker. But those aren't fast attacks. The fast attacks do 11-15 damage. It's not until you have a blue steel axe with a 8 crafting whetstone that it will do 14-18 damage on fast attacks (which averages to 4 damage from special effect). You can get the damage higher sooner by using berserker potions. Also as far as I can tell you don't get multiple bleeds from a single attack, so just in case you had the bright idea to stack crit and use a sword for arterial strikes in the hopes of obtaining up to 3 bleeds on one strike, let me just warn you that you will be disappointed. If you love arterial strikes and crits, just go daggers.
Basically swords are ideal damage boosters for characters with strength below 10, no critical strike skill, no whetstones+berserker potions, and a lot of fast attack spam. So in the end axes hit harder than swords but for most of the game swords are better. This goes back to Quintus from the three guys you can recruit for ants. If you give him the sword you looted last fight, he puts that weapon to good use.