What's more efficient: lightened armor or pocketed armor?

Short story

Pocketed is better for carrying capacity. Lighter Build helps with stealth and AP, but not nearly as much with carrying capacity.

Long Story

The Nukapedia page Fallout 4 Armor Mods has weights and effects listed for various pieces of armor. As of writing this, it was not completely filled out for all items. It does have all the information for metal chest armor, so let's look at that as an example.

Polished Metal Chest Armor weighs 13 lbs.

Lighter Build reduces the weight by 3.2 pounds.

Pocketed increases the weight by 0.8 pounds, but adds 10 carry capacity, for a net gain of 9.2 pounds.

Ultra-Light Build reduces the weight by 6.4 pounds.

Deep Pocketed increases the weight by 0.8 pounds, but adds 20 carry capacity, for a net gain of 19.2 pounds.

So, clearly, pocketed is much better for carry capacity in this case.

Scanning through the other armor, the exact difference between the improvement provided by the pocketed mods and the improvement provided by the lighter build mods varies, but in all cases it appears that the pocketed mods are significantly better for carry weight. This applies to both chest pieces and limb pieces.

Perk Efficiency

You asked whether it would be better to get a higher rank in the Armorer perk or just put more points in strength. That's a pretty complex question, that will depend a lot on subjective preferences. This is because Armorer unlocks a lot more than just the Deep-Pocketed mods, and Strength does more than just increase your carrying capacity.

That being said, if you just want to maximize carrying capacity, you should invest in Armorer.

It looks like Deep-Pocketed only requires Armorer 2 on most, if not all, types of armor. Two points of strength will only gain you 20 lbs of carrying capacity. Putting those two points into Armorer instead allows you to put Deep-Pocketed on all your equipment, gaining you more than 20 lbs on any set of armor.

The short answer:

Pocketed mods (standard or Deep) always out-perform Light or Ultra-Light builds in terms of their net increase to your available Carry Capacity.

However, it is worth noting that the Light and Ultra-Light builds have bonuses and side-effects other than just freeing up Carry Capacity. Particularly, they improve stealth due to the lower total weight in worn Armor and they come with AP Cap boosts.

The long answer:

The boosts that you'll see from these mods is different, depending on the type of armor and the armor slot. The weight of the mods also varies by armor type and slot. So, if you're looking to mix mods within your suit of armor, you'll want to take this into consideration in order to optimize the results according to your play style.

Nukapedia's Fallout 4 Armor Mods page does not yet have complete data on mods across all armor types/slots. However, if patterns for Leather Armor mods hold true across all other armor types, we can assume the following:

• Within an armor type, Pocketed/Deep Pocketed mods provide the same boost no matter what piece of armor they're used on. (e.g.: You get +10 Carry Capacity from Deep Pocketed armor when applied to Leather Armor, regardless of whether it's on the Chest, Arms, or Legs.)
• Within an armor type, the absolute value of a modifier's weight will be less on Non-Chest pieces. (This provides a small boost to the net gain in Carry Capacity from Pocketed/Deep Pocketed, but substantially reduces the weight-reduction value of Light/Ultra-Light.)
• Light/Ultra-Light AP boosts are substantially less (looks like half, rounded down) on Non-Chest pieces.

Given this, if you want to mix modifiers so that you can have an AP boost while still getting substantial gains in your available Carry Capacity, the choice is fairly simple. Put Light/Ultra-Light on your chest piece, and Pocketed/Deep Pocketed on everything else.

That said, as a case example, here's exactly how much a full set of Deep Pocketed Leather Armor will out-perform Ultra-Light Leather Armor in terms of Carry Capacity increase.

Deep Pocketed Leather

• Chest
  • CC boost: 10
  • Weight: .5
  • Net Gain: 9.5
• Non-Chest
  • CC boost: 10
  • Weight: .2
  • Net Gain: 9.8

Total Net Carry Capacity Gain (Chest + (Non-Chest * 4)): 48.7

Ultra-Light Leather

• Chest Weight: -4
• Non-Chest Weight: -1.7

Total Weight Reduction (Chest + (Non-Chest * 4)): 10.8

Essentially, in terms of freeing up room in your Carry Capacity, a full set of Ultra-Light armor can only out-perform a single piece of Deep Pocketed armor - even then, only by a little bit. And it can even still be beat out by only three pieces of Pocketed armor (each having a boost of at least +5, and weighing relatively little in comparison to that boost).

About Stat Boosts for Armorer

Skill Points needed to unlock the various Armorer perks are, in a certain respect, inconsequential to this discussion. All Armorer levels require the same SPECIAL stat level to unlock, with the only other difference in requirement being in overall character level.

That said, currently the highest boost (not counting mod weight) you can get from Deep Pocketed is +20 per piece. To match that boost with your SPECIAL alone would require +10 Strength (+10 Carry Capacity per Strength). If you invest in Strong Back (+50 at level 2), you can reduce the points expenditure to 7.

Armorer 4 requires a character level of 39, but Strong Back 4 only needs level 30. So, anyone who starts with a Strength of 5 or less can get a boost in Carry Capacity, equivalent to a full set of the best Deep Pocketed armor, through SPECIAL and perks by level 30.

(Note: I haven't yet looked into whether Deep Pocketed actually needs Armorer 4, or needs it for all armor types, so it's possible the equivalent boost may be achievable earlier and by characters with a higher starting Strength. If Deep Pocketed is available for all armor types with Armorer 3 or less, then the cost/benefit in terms of skill points per Carry Capacity for each would change substantially.)

Polymer Combat Armor

18.1 pounds with no misc mod

14.9 pounds with Lighter Build mod

11.7 pounds with Ultra-Light Build mod

18.9 pounds with Pocketed or Deep Pocketed mod

Gain 10 pounds carry capacity with Pocketed vs 20 pounds with Deep Pocketed

Each point of Strength gives you plus 10 carry capacity

Long answer short: Deep Pocketed is better than Ultra-Light Build, and equal to two points in Strength per piece of armor with the Deep Pocketed mod when comparing carry capacity alone.