For the first riding-attack it should be the same, 2 jump attacks and 2 vault attacks. But after that, yeah, it does feel like jumping attacks fill up the riding attack meter -if you want to call it that- faster. I can't comment on the SnS and lance jump attacks.
Didn't Capcom say it is based on altitude? Like, jumping from higher above can get you an instant riding attack.
The usefulness of the staff feels like it varies a lot with the terrain and monster speed/attack pattern. If you can get in a lot of vault attacks where it's hard to do jump attacks, thats very useful. If the monster is a bit on the slow side or has long openings between attacks, the staff can provide some dps, because it's not a 'nuke' weapon, the damage is a bit more dependent on getting many hits in.
There are many monsters where something other than a staff is stronger by itself. In a party the beatdowns still make you somewhat useful, but eh, it seems at least not totally game breaking.
Basically what Capcom have added through jumping is ~3 free traps per fight (esp. interesting for the elder dragons), giving you the effects of a run advanced players would usually do (trap/sleep cheese) without the resource drain that getting mats and money for traps brings.
The staff starts out very strong but the other weapons feel like they have a steeper power-curve through raw damage improvement on the weapons and armor to flesh out the playstyle (there aren't really any useful armor skills for staff).
When fighting small and/or fast monsters like the bird wyverns or even Tigarex, it's really hard to hit them, and even if you do, they get out of reach before you can get any real damage in. And landing vault attacks is really hard because of their size/speed. Also Khezu is an ass, always electrifies itself when I vault onto it.
After vaulting into the air, if you see danger/change your mind, press A.
Saved my life a few times.