The Vault-Tec Assisted Targetting System first appeared in Fallout 3, and was both inspired and adapted from the combat mechanics of the original Fallout and Fallout 2. The original games were entirely turn based, which differs greatly from the current generation of first person shooter titles that Bethesda make. A main influence in including VATS was the ability to play Fallout 3 with out any skill in first person shooters, which hardcore fans may not necessarily have had. 
Primarily, this gives you the benefit of a Turn-Based Strategy Approach. However, there are many other benefits to using VATS, both intentional and unintentional. These benefits include Critical Hits, Threat Detection, and the aesthetic Cinematic Combat. There are also V.A.T.S. Specific Perks that provide benefits when particular actions are performed in V.A.T.S., both to improve your combat ability, and include additional function.
Turn-Based Strategy Approach
The primary function of V.A.T.S. is to provide a turn-based approach to combat. In V.A.T.S., Fallout 4 drops into slow motion. During this time, you can choose specific body parts, on specific enemies. Each body part will display the percentage chance to hit, and when highlighted, will reflect the approximate damage you would cause on the enemies health bar.
This allows you to tactically plan out your attack. You can see your exact chances of landing a critical head shot. You can see how much damage your current weapon will land on the target, which gives you a good feel for enemy resistances to specific target areas, or from specific weapons. You should also keep in mind that guns tend to allow targeting specific parts of the enemy, where as melee and thrown weapons will only let you target the enemy.
This can be especially helpful at times where you wish to hit a specific part on the enemy for strategical reason. For example, a good shot to a ghouls leg will blow it straight off. The ghoul might still be at half health, but with the leg gone, the fight becomes a lot easier. As another example, you can target a grenade or explosive, either on an enemies belt or as it is being thrown. This can make for an easy way to dispatch of a group of enemies.
You can perform the same shots in real time, V.A.T.S. just makes it easier. You will also receive a 90% damage resistance, as with previous games. The damage reduction can be hard to make use of, due to the game moving in slow motion, but a quick hand can open V.A.T.S. as your player character is about to take a nasty hit, allowing a fair amount of the damage to absorb into V.A.T.S.
Keep in mind that to rely on V.A.T.S. constantly, you want to favor your agility stat and AP maximization. The higher your agility, the higher the weapon hit chances, and the more AP you have, the more shots you can fire off. AP is more directly impacted by a variety of things, including your weight, and your equipped weapon. There are a few consumables you can take to both replenish and temporarily increase it, and the actions of both running and melee attacking drains it regardless of if you are in V.A.T.S.
V.A.T.S not only lets you slow down time to fight the enemy in a strategic sense, but it also gives you the great benefit of critical hits.
Whenever you hit an enemy in V.A.T.S, your critical hit metre fills up. When your critical metre is full, you can hit the reload button to execute an automatic critical, in V.A.T.S. This guarantees a critical hit, which can be used to strategically blow a limb off a difficult enemy, or perform a high damage head shot at a distance normally difficult without luck and skill.
This ability is directly influenced by Luck. Luck not only determines how quickly you fill the metre, but you can unlock a perk that allows you to "bank" your critical hits. This allows you to execute several critical shots in one go, where normally you can only hold a single critical shot at a time.
This tactic comes from previous Fallout titles, and may reduce the experience of the game as it was intended.
V.A.T.S is designed to allow you to assess the enemy, and attack accordingly. As a result, V.A.T.S will detect targets that you normally would not see.
When you are in a dangerous area, you can attempt to enter V.A.T.S to locate potential threats. Enemies that are visible, but far in the distance or partially obscured by cover, will be highlighted. So will land mines. As a result, you can be more aware of the enemy presence, and reduce the risk of being accidentally blown up by land mines. In actual fact, this is a very important feature, if you find yourself easily blown up by unnoticed mines.
This is also a useful tactic, when approaching settlements or wandering NPCs, to determine whether they are aggressive or not. Traditionally, a friendly target will be displayed in a green outline, where a hostile target will be outlined in red.
Providing a great sense of cinematic immersion, actions carried out in V.A.T.S. will be acted out in slow motion. Furthermore, the camera will move to different angles, to give you a better view. It will zoom in on critical hits, and will even follow the enemy, at times. In some situations, this can help you determine the actions of enemies, where they might normally be too far away to tell.
V.A.T.S. Specific Perks
There are several perks in the game that either add function to V.A.T.S., or provide increased ability in its use. Please note that accumulative benefits only count for the shots performed within the same sequence. To ensure I noted everything, I have used the Fallout 4 Wikia, for its page on Fallout 4 perks.
- Iron Fist
- (Rank 5, Strength Rank 1, Level 46): Critical hits with melee weapons are guaranteed to paralyze the target. This means that even if you do not kill the enemy, the enemy will not be able to move or attack back after the attack.
- (Perception Rank 3): Along with displaying the targets health, damage resistances will also be displayed in V.A.T.S. to help you determine the best weapon to target the enemy with.
- Demolition Expert
- (Rank 4, Perception Rank 5, Level 34): Explosives shot in V.A.T.S. explode for twice the damage. This can make shooting an enemies grenade out of their hand especially useful.
- (Rank 3, Perception Rank 8, Level 26): Non-automatic scoped rifles have an extra 25% chance to perform head shots in V.A.T.S. making it much easier to perform high-damage shots, if you are using a sniper rifle.
- (Rank 1, Perception Rank 9): Enemy body parts can be targeted from V.A.T.S., even if they are behind cover. While you have a decreased hit chance, this means you an potentially perform head shots on enemies that are in cover.
- (Rank 2, Perception Rank 9, Level 28): Upgrading this perk removes the accuracy penalty, making these shots just as deadly as regular line of sight shots.
- Concentrated Fire
- (Rank 1, Perception Rank 10): Each consecutive hit on the same targeted body part gains a +10% to accuracy, in V.A.T.S. Obviously the more shots you can get off at once, the greater potential this perk can grant. But even with low agility and accuracy, this perk can back some much better odds at hitting a critical target.
- (Rank 2, Perception Rank 10, Level 26): Increases the accuracy bonus to +15% per hit.
- (Rank 3, Perception Rank 10, Level 50): Increases the accuracy bonus to +20% per hit, and adds an increase of +20% to damage, to sweeten the pot.
- (Intelligence Rank 1): Holding down the V.A.T.S. button will display an in-game way point marker to the closest quest objective.
- Quick Hands
- (Rank 2, Agility Rank 8, Level 28): Reloading does not cost AP. This means you do not suffer a penalty, and potentially lose a shot, if you have to reload your gun half way through performing your V.A.T.S. shots.
- (Rank 1, Agility Rank 9): Significantly increases the range of melee attacks in V.A.T.S. allowing you to perform melee attacks without having to be as close to the enemy.
- (Rank 2, Agility Rank 9, Level 29): Further increases the range of melee attacks in V.A.T.S., while increasing melee damage with greater distance.
- Gun Fu
- (Rank 1, Agility Rank 10): Every shot past the first will deal another 25% damage.
- (Rank 2, Agility Rank 10, Level 26): While the second shot will still receive a +25% to damage, every shot past the second will instead receive +50% to damage.
- (Rank 3, Agility Rank 10, Level 50): Any shot past the third shot will automatically deal critical hits.
- Mysterious Stranger
- (Rank 1, Luck Rank 4): A Mysterious Stranger will appear at random, to assist you in V.A.T.S. combat.
- (Rank 2, Luck Rank 4, Level 22): Increased chance of the Mysterious Stranger appearing in V.A.T.S.
- (Rank 3, Luck Rank 4): Increases the chance of the Mysterious Stranger appearing in V.A.T.S., but also adds a chance for Mysterious Stranger kills to fill your Critical Hit metre.
- Critical Banker
- (Rank 1, Luck Rank 7): You can now store two critical hits, instead of one.
- (Rank 2, Luck Rank 7, Level 17): You can now store three critical hits, instead of two.
- (Rank 3, Luck Rank 7, Level 43): Not only can you store four critical hits, but filling the critical hit metre has a chance of rewarding two critical hits at once.
- Grim Reaper's Sprint
- (Rank 1, Luck Rank 8): Any kill in V.A.T.S. has a 15% chance to restore all action points.
- (Rank 2, Luck Rank 8, Level 19): Any kill in V.A.T.S. now has a 25% chance to restore all action points.
- (Rank 3, Luck Rank 8, Level 46): Any kill in V.A.T.S. now has a 35% chance to restore all action points, and may also fill the critical hit metre.
- Four Leaf Clover
- (Rank 1, Luck Rank 9): Each hit in V.A.T.S. has a chance of refilling the critical hit metre.
- (Rank 2, Luck Rank 9, Level 32): Each hit in V.A.T.S. has an increased chance of refilling the critical hit metre.
- (Rank 3, Luck Rank 9, Level 48): Each hit in V.A.T.S. has an even further increased chance of refilling the critical hit metre.
- (Rank 3, Luck Rank 10, Level 50): Any ranged attack may ricochet back at the enemy, with a chance of killing them. If this shot kills the enemy, there is a chance it will completely refill the critical hit metre.
Killshot (Companion Perk, Robert MacCready): Increases the accuracy ,when targeting enemy heads in V.A.T.S., by 20%.