SHOUT OUT TO : u/Zeiin
The equation for damage, as far as we know, is:
We make a few notes here just by looking at the equation:

Pierce affects a denominator, making it naturally the largest scaling modifier up until softcap. The more pierce you have, the more damage you get from additional sources of pierce. To clarify, 3% pierce is roughly 3% damage, 20% is 25% damage, and 30% is 42% damage.

Pierce also doesn’t rely on enemy defense, it is a standalone modifier that does act as a fractional multiplier against enemy defense, causing it to be more effective vs higher defense enemies, but against enemies with NO defense, Piercing is equal to a %damage increase.

VS dummies, that have no defense, 20% piercing and 25% increased damage are the same.

Total damage is applied to a numerator, meaning it gets divided by every other piece of damage mitigation.

Physical piercing directly reduces a source of damage mitigation, and thus has a hard cap, namely when (15 * Physical Piercing) == Enemy Resistance, making the last part of the equation:

(1500 – 0) / 1500 = 1, effectively nullifying the resistance stat as a whole.

Due to this hardcap, the scaling of physical piercing gets turned into: “Get enough physical piercing to nullify this damage mitigation”.
One takeaways from how Pierce and Physical Pierce then work, is that both stats not only affect your Base damage, but they also make your %damage increases more and more effective.
The reason this is the case, I’ll explain with a hypothetical.
Now let’s look at the choosing a physical piercing roll or a total damage roll on your weapon: If an enemy has 100 defense, and 100 resistance, and you have no piercing or physical piercing, and deal 100000 damage per hit as a base.
Total Damage = 100000 * (1/100) * (1/1) * (1400/1500) = 933.33 Damage dealt.
Let’s say we put on a weapon with 10% physical piercing.
Total Damage = 100000 * (1/100) * (1/1) * (1500 – (100 – (15 * 10)))/1500
This simplifies to:
Total Damage = 100000 * (1/100) * (1/1) * (1500/1500) = 1000 Damage dealt. <Phys pierce roll
Let’s say instead, we put on a weapon with 3% total damage instead.
Total Damage = (100000 * (1+0.03)) * (1/100) * (1/1) * (1500 – (100 – (15 * 0)))/1500
This simplifies to:
Total Damage = 103000 * (1/100) * (1/1) * (1400/1500) = 961.333333 Damage dealt.
Choosing the physical piercing here gives us 4% more damage than choosing total damage.
Keep in mind though, if you already have physical piercing, taking any extra physical piercing will add nothing to your damage because physical piercing can’t turn resistance negative. It just nullifies it.
So once you reach cap of 16.6% physical piercing to entirely nullify chaos raid boss resistance, you don’t benefit from any extra.
Let’s now take a look at overall scaling.
The point I asserted earlier is that % damage increases are more and more effective the more pierce and physical piercing (up to cap) you have.
Let’s now consider the hypothetical:
If an enemy has 100 defense, and 120 resistance, and you have no piercing nor physical piercing, and deal 100000 damage per hit as a base.
Total Damage = 100000 * (1/100) * (1/1) * (1500(120(15*0)))/1500
Simplified this is:
Total Damage = 100000 * (1/100) * (1/1) * (1380/1500) = 920 Damage Dealt.
Now let’s say you add 20% increased damage from various sources to your gear. This can include boss damage, total damage, dark damage, melee damage, etc.
Total Damage = (100000*(1+0.2)) * (1/100) * (1/1) * (1500(120(15*0)))/1500
Simplified this is:
Total Damage = 120000 * (1/100) * (1/1) * (1380/1500) = 1104 Damage Dealt.
This is a flat increase of 184 damage.
Now let’s say we start off with no % Increased damage, and we just have 20% piercing and 10% physical piercing.
Total Damage = 100000 * (1/100) * (1/(10.2)) * (1500(120(15*10)))/1500
Simplified this is:
Total Damage = 100000 * (1/100) * (1/0.8) * (1500)/1500 = 1250 Damage Dealt.
Now, from this point, let’s add 20% increased damage from various sources to your gear.
Total Damage = (100000 * (1+0.2)) * (1/100) * (1/0.8) * (1500)/1500
Simplified this is:
Total Damage = 120000 * (1/100) * (1/0.8) * (1500)/1500 = 1500 Damage dealt.
This is a flat increase of 250 damage.
Conclusions to draw from this:

If you lack piercing, you prioritize it over any other stat, until the 30% soft cap.

If you lack physical piercing, you prioritize it over any other damage increase besides flat piercing.

If you have physical piercing at cap, you prioritize it less than any other stat and always choose % Increased Damage.
tl;dr Pierce > Phys pierce (if not hard capped) > Boss Damage >= Total/Melee/Ranged Damage >= Any Other Damage >= Phys pierce (if hard capped)
Boss damage > Total damage only because boss damage rolls much higher.
Source for initial damage calc: http://www.inven.co.kr/board/maple2/4276/5231