The old Fallout games have a bit similar AP based turn-based combat with Snapshot and Free Aim, but costs different amount of AP
So here comes my question, why there is no AP difference in PP?
In the PP, the free aim provides fundamental advantage, and I do not see any purpose to use a snapshot
Snapshot tracks with the target, whereas free aim does not. It you just need to put damage into something, snapshot can be the better choice.
Here are some advices why and when to use snapshot instead of free aim:
http://wiki.phoenixpoint.com/Combat#Free_aim_vs_snapshot
In short: for burst weapons like assault rifles and with a free line of fire snapshot is often the better choice because it follows the moving of the target.
Not really - unlike Fallout1&2, there is no inherent advantage when using one over the other. The difference between automatic and free aim is that the first one aim at the target of your choice, while the other at a specific point in space.
You will most likely want to use free aim a lot to prioritise squishy parts of the target or maximise your chance to land a shot. I also find that free aim gives me better information of my hit chances then quick aim.
However, freeaim is unreliable when it comes to burst weapons. Bullets will be projected over certain amount of time, while the enemy will be moving - passive animation but also ore extreme reaction to getting hit. That means that enemy can move out of for the way and dodge a big chunk of damage if you use freeaim with burst weapons. Of course, one can learn enemy animations and adjust accordingly, but auto aim locks to enemy centre of mass - as they move your aim will adjust as well, and give a decent chance of hitting squishy part as well. I also find it useful for small targets like worms or mind fraggers.
As such both aiming modes are tools to manage the projectile simulation system, rather then AP investment vs better aim mechanic. Giving freeaim extra cost would have also some unfortunate consequences - like shooting a wall would for some reason take more AP then aiming at the enemy hiding behind cover.
Simply, you would shoot to much 
From what I’ve seen, free aim isn’t as reliable when you’re aiming at something small and fast-moving (like a Triton arm), since the target will continue to move while you take your shot. It helps to watch and lead the target, but it’s not foolproof. Snapshot just defaults to center mass (I think).
So, what happens if you give the Quick Shot skill to all weapons?
1 +2 +3Examples:
Early PX weapons
Heavy Weapon
Exampl2: Pistol = 0,5AP, 1 bullet, -60% acc … …
EDIT
Snap Shot is a shot almost without hesitation, on reflexes.
Free Aim spends time analyzing and counting.
I like the main concept of having different shooting possibilities for all weapons and would go even a step ahead and think they could get even 3 modi, 1AP-2AP-3AP.
Generalized:
1AP weapons:
2AP weapons:
3AP weapons:
3AP is superfluous, because “1AP х 3” is more profitable without losses
1AP is an unattainable limit to the speed of human reaction
Even when I think it can be balanced and also explained you’re maybe right, I’m just dreaming from the good old days with the original xcoms or JA2 where they have done so much more with the time units 
This is the Spirit of the Concept above. But now I see a problem, there is no place for skills.
More precisely, there is no way to use skills that further reduce AP.
EDIT
But it will be possible to remove the Quick Aim sniper skill. 
Or rather, replace it with the Sniper skill for Accuracy (active) or Pistol skill (as support).
This is generally a problem with almost any weapon change. Also all the damage buff abilities (Marked for Death, Sneak Attack, the proficiency perks …) could scramble anything around.
Edit:
Best example are the “weak” Ares and Daimos ARs, they change into murder machines when combined with the double damage from Sneak Attack.
I have trouble finding a good function for multiple weapons firing mods - alowing to have better accuracy via time investment essencially means weapons costing more action point depending on distance of engagement. Naturally, the further you are the better deal it is to invest more action points to reliably land a shot. On the other hand, there is little benefit of investing into aim boost if one is close.
“But UFO had it and it was 96!” I hear you say. Yes, but UFO had little to no weapon variety. Same rifles acted as sniper/automatic/low accuracy single shot weapons. This is covered already, and with better balance and nuance via multiple weapon types. We have high AP, single shot long range sniper rifles, 2AP burst assault rifles and 1 AP, low damage SMGs. That’s pretty much the same thing, just achieved in a different way. Bigger weapon varity, and ability to equip multiple weapons, essencially overrides UFO’s aim system.
If there is a way to adapt the system for Phoenix Point’s need, and give it new function that would add to the game in a meaningful way - I don’t see it.
Fair enough and you’re probably right, sometimes I go to far with my thoughts. In my opinion, generally the existing system we have with PP isn’t bad, we have a form of automatic that aims at the centre of mass and follows the movement, what is good for burst weapons, and we have the manual aim to be more precisely (but not more accurate) to aim specific body parts or just fire on some cover/walls/windows. Anything above that would maybe provide more flexibility but probably also many problems to balancing.