Weapon EXP is different from normal EXP in that it goes towards increasing a character's Weapon Rank, allowing them to use more powerful weapons. Staves have especially high ranks in this area, but most weapons are only 1 or 2 in this particular stat (except in Fire Emblem: Radiant Dawn). The general rule that the harder a weapon is to use (based on things such as Hit%), the higher Weapon EXP it will have.
It is unknown why SS-ranked weapons have points in this category, as that is the highest rank obtainable and it no longer matters.
Myths
Some people say Forging heavier weapons in Path of radiance/Radiant dawn will get you more experience. This is not true. Examples are that the Silencer gives 2/3 more WEXP then Steel Longbow, but Silencer is 1/3 of Steel Longbow's weight. Rolf's Bow gives twice as much WEXP as an Iron Bow. Cymbeline gives 8 WEXP but it is the same weight as Elfire. The Silver Axe gives 5 WEXP but a Steel Poleax gives 4 WEXP, not to mention that the latter has an extra 6 weight. Mend and Torch give the same amount of WEXP yet Torch is 75% heavier. Therefore the 'heavier weapons give more WEXP' is a belief only, since the heavy Devil Axe gives 8 WEXP.