Axe
The very prospect of a not-so-sharp wedge of iron decending forcibly on the the back of one's neck, perhaps more than once, evokes shudders of horror. Yet this method of execution , rather than that of hanging, was actually granted as a privilege to those of noble birth, death by cold steel being considered more honourable , akin to being slain on the battlefield.
The execution axe itself was not unlike the battleaxes used in combat which , far from being finely honed and balanced weapons, were designed solely to batter through armour and cleave through helmets. Likewise, the 'heading axe' , as it was called, was little more than a blunt, primitive chopper which crushed its way through the flesh and vertebrae of the victim as he or she, knelt over the block on which their head lay.
One thing was sure, the executioners axe did not always cut through on the first strike nor did it ever hit its target accuratley.
Sword
Unlike the axe, the sword was a little bit more reliable in accuracy along with killing. And unlike the axe it could be thrusted into various spots and positions, like in the movie "Gladiator" when the soldier is about to thrust the sword into Maximus's spine in Germania.
in the swords case, it was not usually used in formal and/or public executions, they got there use mostly on the battlefield or away from cities where swords were plentifully strewn across bloody grounds. The picture above is a rare use of the sword in execution, the picture above is a French executioner.
A typical sword of execution would have a two inch (5 cm) wide blade which was blunt ended and measured about 40 inches (100 cm) in length. One example from Germany bears and inscription which translates: "Whenever I raise my sword I wish the sinner everlasting life."