Swords work well for "on-demand" effects. For example, while an axe would have to charge adrenaline to use disrupting chop, sword's savage slash would interrupt a spellcaster (or otherwise) immediately, if within melee range. The same applies for hamstring over axe rake.
Also, sword has an easy time applying all of its conditions in rapid succession (perhaps not in theory, but in practice) with hamstring, sever, and gash one after another.
In this way, swords are better against single enemies, such as PvE bosses, or enemy players, where they can apply their conditions and use their very tightly controlled skills to their advantage.
However, most sword skills require energy, and with a warrior's meager energy pool and regeneration, energy denial can easily disable a sword warrior.
Axes work well for pure damage. The increased chance for critical hits with higher axe mastery makes the normal damage above and beyond what a sword would do. Also, its various adrenal damage skills like cleave and executioner's strike make for easily usable (especially with an adrenaline increase of some kind, as with battle rage or berserker's stance) for higher damage along those lines as well.
Since axes don't truly rely on energy all that much, they are able to continuously hit for very powerful adrenaline attacks--this combine with many options for deep wound makes damage come very quickly for an axe.
The problem lies, of course, in the fact that it requires adrenaline to charge and something like sympathetic visage and the like that disable adrenaline can negate an axe warrior. However, fewer enemies carry anti-adrenaline, compared to the number that carry anti-energy. Usually, though, this adrenaline reliant attacking is an advantage rather than a disadvantage, especially with, as said, faster adrenaline charging.
As a final note, regarding mostly PvE, axes are better than sword against multiple enemies with just cyclone axe. Even beyond that though, axes with something like For Great Justice quickly charge and quickly eliminate opposition--and even without such a skill, enemies are plentiful in PvE to charge from.
Finally, hammer is used primarily in PvP at this point due to its knockdown ability. Although it attacks slower than the above two, which makes Strength of Honor and the conjures less useful (Monk and Elementalist secondaries, respectively) its primary goal is denial of action, which like sword, works far better against single enemies.
However, Earthshaker and Aftershock have worked excellently when I tried it. The problem is, of course, it isn't a combination that you can trigger all that often with Earthshaker's high adrenaline cost, which is more troublesome for a hammer due to its slower attack. Of course, that was a test and it did work for some damage, if not as effective as axe damage.
Overall, although I have some experience with hammers, I don't have nearly as much as I do with axes and swords. Though from what I've seen from them, hammers also rely on adrenaline highly like, if not more, than axes.
