Where in the core books do they go into detail the differences between personal and vehicle scale weapons? Do they at all?

AoR pg 236, the giant box that takes up the bottom half of the page.

Short version: Vehicle scale weapons are devastating against personal scale targets, and engaging armored vehicles is not something a footsoldier should try to just improvise on the fly.

I will hook my question to this thread.

During a chase or a simple planetary vehicle combat, how would handle melee attacks or other actions like using Athletics to jump from one speeder to an other one?

Assuming the chase mechanic continues even after getting that close (which they can, but they usually end or switch to normal rounds at close or engaged range depending on the scale) you'd run it normally. Remember the chase mechanic occurs, and then you run the round normally, the chase check doesn't count against the actions a character can take once the round starts going.

I will hook my question to this thread.

During a chase or a simple planetary vehicle combat, how would handle melee attacks or other actions like using Athletics to jump from one speeder to an other one?

Personal Scale engagements can happen in chases when the vehicles are within close range of each other. The RAW tells us GMs to arbitrate how close vehicles are to each other in Personal Scale (again, they'd have to be at close range in Vehicle Scale), and then I'd add 2 Setbacks for the "unstable terrain" on any Melee or Athletics checks.

Triumphs on an Athletics check to jump from vehicle to vehicle could be use to cause a critical injury with your Melee weapon, or something like that. You could have your friend at the helm perform a Piloting check before you your turn comes up, to try and make it easier for you to take the leap (either by closing distance, or gaining the advantage, or maybe even trying to actually engage with the other speeder, side-swiping it or locking frames or something..