A previous response has some great answers to your question.

Each weapon had an advantage in certain circumstances but would be ineffective in others. An axe most effective against plate armor and chainmail. A sword is most effective against unarmored or lightly armored opponents. This is because axes have a heavy head which generates momentum when swung. Swords slice or stab but these motions may be deflected off of armor.

If you are hit with an axe even if your armor manages to stop the head from penetrating your chest your armor is likely damaged and you will probably feel like somebody just took a sledgehammer to your chest.

A knight would have an assortment of arms for use in different situations. If he expected to be part of a cavalry charge against archers he might use the sword to slice as he passed. If he expected to be part of a heavy cavalry charge against infantry he might use a lance. If he expected close combat against armored opponents he might choose an axe.

For average soldiers axes were handy because they served multiple roles. The English Archers at Agincourt used hatchets to make a defensive line of stakes to protect against the French Cavalry charge. Later when nearly all of their arrows were spent the English archers attacked the worn out French troops with those hatchets.