The basic tools - axe, pickaxe etc. - differ from weapons simply because they specialise in the aquisition of materials from non aggressive sources, whereas weapons specialise at dealing damage. Tools can still be used to attack enemies in the same way that weapons can still mine things, to an extent, but tools each have a primary use as listed below.
Other tools simply have practical uses. Either way, the following are all necessary items for the Minecraft player and have very useful reasons to exist.