According to Tom Gaylord, (not me; I havnt used them before)
"Weaver mounts don't work on airguns
Weaver mounts originated just before 1950. They were the first attempt to provide standardized scope mounts for rifles. They feature an integral recoil block to keep the rings in one place. Before Weaver mounts, rifles were most often drilled and tapped for small setscrews that sometimes snapped under heavy recoil. The Weaver base has a 3.8mm cross-key slot that accepts special Weaver keys located on the bottom of all Weaver rings. Weaver bases are great for firearms but too wide for air gun rings. Also, the locations of the cross slots are not standardized, so Weaver bases work best with two-piece rings."
I like the Picatinny because they have exact measure slots, so you can interchange rings and not be afraid of them not matching perfectly. Because Weaver does not have exact measurements, the slots might not fit certain rings.