An axe holds two main advantages over a sword: (a) it is cheaper and easier to make, as only the axehead is made from metal, and (b) it can deliver a blow with greater force, which may be desirable if the foe is armoured. It may also be avaiable as a tool in times of need, although axes that were designed for combat differ substantially from the tools of a lumberjack or a carpenter.

The main disadvantages is that an axe is top-heavy, which makes it more cumbersome and tiring to use. A well-balanced sword is a much nimbler weapon. For this reason and the fact that the handle of an axe is made from wood, an axe can not be used for fencing, while a sword can.

These are general explanations. For example, there are sub-types of swords that are designed to be top-heavy (i.e. falcata) and are therefore less useful in fencing.

A war-hammer is a specialised weapon designed to pierce heavy armor and to deliver blunt force. It came into use as more and more heavy armour appeared on the battlefields of Europe. It was used by footmen and mounted warriors alike, with the main difference being the length of the handle.

To answer the question: the sword never made other forms of melee weapons obsolete. Other forms of weapons offered benefits that the sword was not able to match, like reach or armour-piercing capabilities.