From what I can see, the three skill branches seem to favour different approaches to combat.
In addition to the extra types of attacks you gain for each weapon type (e.g., Chop for Axes, Bash for Maces, Slash for Swords), you get increases to different attributes at different skill levels, which will allow you to hone your fighter as you want:
- Axes give increases to Strength and Health
- Maces give increases to Vitality and Health
- Swords give increases to Strength, Health, Energy and Dexterity (each to a necessarily lesser degree than the other weapon skills)
So in essence, if you want a pure brute, unstoppable in close combat, Axes may be a good choice. If you want more of a tank build - to be able to withstand more attacks - go for Maces, as the increased Vitality will let you level up your Health even faster. Swords seem to be the all-rounder, with a mix of upped combat strength as well as accuracy and energy.
Source: Just the descriptions of the skills in the game menu.