Weapon EXP (abbreviated WEx or WEXP) is different from normal EXP in that it goes towards increasing a character's Weapon Rank, allowing them to use more powerful weapons. Staves in particular give a lot of weapon experience, but most weapons only give 1 or 2 weapon experience (except in Fire Emblem: Radiant Dawn). Weapon experience is always 1 in both Fire Emblem: Thracia 776 and Fire Emblem: Binding Blade (except for staves), and is always 2 in Fire Emblem: Shadow Dragon (except for the real Imhullu and Dragonstones, where it is 0). In FE7-10, different weapons and staves have varying values of weapon experience (1-8).
Seemingly, the trend is that the harder a weapon is to use (based on things such as Hit%) or the more dangerous, like the Devil Axe in most games, the higher weapon experience it will grant to the user.
It is not known why SS-ranked weapons have points in this category, as that is the highest rank obtainable and accruing weapon experience no longer matters.
It was previously thought that Forging heavier weapons in Path of Radiance/Radiant Dawn will grant their users more experience. This is not true. Examples are that the Silencer gives 2/3 more weapon experience than the Steel Longbow, but Silencer is 1/3 of Steel Longbow's weight. Rolf's Bow gives twice as much weapon experience as an Iron Bow. Cymbeline gives 8 weapon experience but it is the same weight as Elfire. The Silver Axe gives 5 weapon experience, but a Steel Poleax (which has 6 more weight than a Silver Axe) gives 4 weapon experience. Mend and Torch give the same amount of weapon experience yet Torch is 75% heavier. Therefore the 'heavier weapons give more WEXP' is a belief only.