TL;DR the TL;DR: Skull Piercer is +25% (not 40% as the title claims, sorry!). Helmet reduction is 10%/20%/25%/25% of TOTAL damage on a head shot for each tier.
 Seriously THANK YOU everyone for helping me/us solve this mystery. I'm 99.999999% sure this is the correct math, now.
TL;DR:
Chad Armstrong (Respawn Dev) has deleted his original helmet reduction tweet (the source for all the wikis and other websites for these numbers), because it's wrong. Helmets reduce TOTAL damage of head shots by 10/20/25%, not bonus damage by those previously quoted numbers. A head shot is still preferred to a body shot, even a 1.5x against a tier 3/4 helmet. If you can hit them, at least. Obviously 1.0x is more than 0x. ;)
Also, as mentioned in the title, Skullpiercer adds 40% extra damage to head shots. At least for the Wingman.
 Longbow Skullpiercer is apparently +25% head shot damage. See below for more info.
EDIT: Hold up. We are getting conflicting numbers! I'm seeing 84 damage head shots with image proof. Chad might still be incorrect, at least as far as the purple helmet goes.
SECOND EDIT: As I'm watching streams today I'm seeing totally different numbers. I think the devs shadownerfed skullpiercer to 25% for both Wingman and Longbow (since the bloke that came up with the LB #'s got it today as well). The damage is now in line with 25% for Skull Piercer on the Wingman:
90 * 1.25 (SP mod) = 112.5 * 0.75 (epic helmet) = 84.375. People are seeing 84 damage now.

So damage numbers for Blue/Common are now 112.5 * 0.8 = 90 for blue (which we see now), and 101 for common (which we also see now).
 Unless I'm certified crazy and these have been the numbers all along. But I specifically watched a game where the numbers were higher! And yet 113 and 101 still lines up for no helmet and white helmet...so...I might be legit stupid, boys! Apologies! 25% SP mod for both is the answer, though. 10/20/25% on TOTAL is also the answer.
 Goddammit I hate when the WRONG MATH lines up! 25% is the answer. Sorry!
All sorts of wikis and gaming sites like PC Gamer have used this tweet from Chad Armstrong to represent the helmet reduction numbers.
I've been trying to do the math with those numbers and it never worked out  particularly with trying to figure out Skullpiercer damage. I was watching Dizzy's stream since he likes to use the Wingman, and noticed two different head shot values (101 and 113) when he had Skullpiercer equipped. I tried to math it and nothing came out correct.
Then I saw this post randomly while browsing the subreddit, and everything clicked.
So here's the tweets /u/TheDunadan mentions, just so you don't have to click on a link to find another link (I just wanted to give credit to the guy that helped me, thanks dood):
Tweet 1  He clarifies that the numbers are wrong.

Tweet 2 in response to Centrip Studios (no idea who this guy is) he gives the actual numbers. It's 10/20/25% on TOTAL head shot damage.
 This modifies head shot damage to 180%/160%/150% for 2x multipliers, and 135%/120%/112.5% for 1.5x multipliers.
 So you can see that even the "worst case" scenario of a 1.5x weapon and a purple helmet, it's still more damage on a headshot, despite it modifying total damage instead of bonus, as we previously thought.
The math works out for the Wingman numbers I was seeing if Skullpiercer has a 25% increased multiplier to head shot damage. Here's the math for those that care:
90 * 125% = 112.5 for a Wingman head shot with Skullpiercer and no helmet. Rounded up to 113.
112.5 * 90% = 101.25 damage for a white helmet. Rounded down to 101.
112.5 * 80% = 90 damage for a blue helmet. This is exact.
112.5 * 75% = 84.375 damage for a purple helmet Rounded down to 84.
The numbers for no Skullpiercer should be 90, 81, 72, and 67.5 per helmet tier. This checks out with my memory as well. I know I've seen 81 and 72, at least. ;)
What Does This Mean?
It means that white and blue helmet are really good, and purple helmet isn't really as much of a priority as purple armor is. Yes a max helmet will protect you from a Kraber head shot (The incorrect math will give you the same answer as the correct math  187 damage  this is why there is so much confusion), and 5% extra head shot reduction likely adds up to more survival against some weapons, but it's not as big of a deal as finding a white helmet and upgrading it to blue. Certainly pick it up if you find it, but don't, like, go far outside the ring to pick one up, I would say. And certainly don't spend the time looting a death box for one if you're not safe.
 Then again, a purple helmet will save you from a 2headshot Wingman as well, so maybe it is important to grab, since Wingman is so meta right now? Discuss below.
Purple armor on the other hand, you probably still want to do this. "Refreshing" my armor using a death box that was conveniently behind cover has saved me a couple of times. Use your brain on this one, obviously don't loot while your teammates are dying, I'm not saying that.
Also it just flat out means stop quoting the wrong information. Correct others who quote the wrong information as well, please. And, uh, do it nicely. ;)
Another thing to note is that Skullpiercer might add a different value than 40% to Longbow head shots, I have no idea, as I haven't taken time to research. Very few streamers are 'actively' playing the sniper role. Kephrii does, he's the only one I know of.
See this post below by /u/Enchantz7. Apparently the Longbow Skullpiercer head shot damage boost is 25%. :) So they are different.
EDIT: I'm a dumbass. It's 25% for both, not 40% for the Wingman. I was assuming 113 damage was on a common helmet but it was with NO helmet. The math worked perfect except with epic helmets, thanks for those that provided picture evidence, it's sorted out.
EDIT  Proposed Change to Wingman
With the numbers being officially "discovered" now, I'm thinking the Wingman and Longbow should honestly 'swap' their skullpiercer modifications. This would bring Wingman headshots down significantly enough that getting twoshot isn't a meme anymore (need 3 headshots with blue+ helmet, which should be way more rare). Also another user mentioned something that got me thinking  the projectile speed should be reduced such that firing from midtolong range shouldn't be as effective as it currently is.
Nerfing the range of the Wingman as well as its headshot damage should appease the community without hitting the gun so hard it's the next Mozambique. Also I think the community in general thinks Sniper Rifles could use a bump in turn.
See second edit  Wingman is at 25% already. Whoops! I stand by adjusting the bullet speed to nerf its range. It should be a primarily shortrange powerhouse (it's a heavyammo pistol, those have always had shortrange advantage since the 007 days), and still have some kick at mid range, but be inferior to other options. As it is now it's even strong longrange, and that's dumb.
SECOND EDIT  Big Clarification
Apologies, guys. I've announced it everywhere above, but I made a mistake. Skull Piercer is +25% damage, not +40%. I assumed the wrong tiers of helmets and the math still "worked" to give me, essentially, the wrong answer. See math and discussion above for details.
I've edited the relevant bits in the main post to reflect the correct math, while leaving some of the verbiage crossed out like this for incorrect stuff I felt was important to keep but also highlight as incorrect.