The exception to that, of course, is the throwing knife. A knife with the blade heavier is usually a dagger. The exception to that, of course, are specialised chef's knives (chef's knife, cleaver, etc) which take their form from the Axe family.
Swords have one of two functions. To stab, or to slash. The rapier, foil and broadsword were used to stab. They're evolved from the spear family of weapons (which itself evolved from little more than a dagger tied to the end of a staff).The Bastard sword, Long sword, Two-handed sword, Cutlass, Falchion, Claymore, etc were more specialized axe heads on absurdly short handles. Meant for slashing and chopping. While all swords (for the most part) had both points and edges, stabbing swords lacked the weight to penetrate anything heavier than a linen shirt, and slashing swords were too damn heavy to stab with effectively. Curved swords like the cutlass and scimtar were even worse, as their point usually wasn't even in line with the centre of mass in the first place.