Slashing weapons, like axes, are typically most effective against relatively small creatures that have weaker armor than you do weapons. They'll lop off limbs efficiently against such creatures, thus rapidly and drastically reducing the enemy's ability to counterattack. Their ability to take off limbs also makes them most effective against inorganic creatures, such as undead or some forgotten beasts, as the other damage types don't do much against such things. Unfortunately, against creatures that are too large, they'll be unable to do more than just damage limbs or musculature and will thus take a long time to kill big things without seriously reducing their fighting power.
Piercing weapons, like spears, are typically most effective against very large unarmored organic creatures. They'll damage organs directly and can thus actually kill such enemies more quickly than an axe, which would likely only damage limbs. The trouble is, they're not really better at piercing armor than slashing weapons, and if an enemy is small enough that your attacks can remove limbs altogether the difference between a piercing attack that kills an enemy relatively quickly but leaves it fighting at 100% until it drops and a slashing attack that kills slowly but reduces fighting power in the meantime favors the slashing weapon.
Swords are somewhat like weaker axes in terms of cutting power, but also have the piercing power of weak spears. They're thus versatile weapons usable against anything.
Blunt weapons are most effective against armored targets. Their attacks can bypass armor far more effectively than slashing or piercing weapons, allowing them to score instant kills where slashing or piercing weapons wouldn't. Unfortunately, their overall effectiveness is relatively low; the damage they cause to limbs and organs with a blow that bypasses armor is less than another weapon would cause with an armor-bypassing blow, so where there's no armor to bypass they're worse. Against large creatures, they'll have much the same problems as slashing weapons: ability to damage musculature, but not to actually kill without a large strength, skill, quality, and/or material advantage.
Picks are believed to be very good weapons, as they behave something like slashing weapons and something like piercing weapons without really being either. They can effectively sever appendages, but also can penetrate to damage organs.
Overall, I myself typically equip all of my soldiers with axes. High-quality bronze in a legend's hands is enough to kill forgotten beasts (the largest common enemy) efficiently, while steel or, better yet, adamantine can allow even a middling-strength novice to shear off anything's limbs. That material advantage means that the sizes against which axes are most effective go up to be larger than even the largest creatures, so they're simply excellent all-around weapons.