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Breakdown

The most important stats to consider for overall weapon damage are: Skill, Strength and Body.
The differences between the various letters are relatively small (see mechanics). The most important thing to consider is if a Weapon scales with a Stat or not.
Skill adds B+ damage to Dual Swords, B+ damage to Rifles, C damage to Spears, C+ damage to Kusarigamas, C+ or C damage to Swords C+ damage to Bows.
Strength adds B+ damage to Axes, C or D+ damage to Swords, C+ or D+ damage to Dual Swords, C damage to Spears, and C+ damage to Hand Cannons.
Body adds B+ damage to Spears, C+ or D+ damage to Axes, D+ to Kusarigamas and C+ damage to Rifles. (DLC adds C+ for Tonfas)
Dexterity adds B+ damage to Kusarigamas, but adds nothing to other melee weapons. It does add D+ damage to all ranged weapons.
Heart adds B+ damage to Swords and Bows, and C+ damage to one half of Dual Swords, and a D+ to the other half. (DLC adds C+ for Odachis)
Stamina only ever adds B+ damage to Hand Cannons, C+ damage to Axes and D+ damage to Hammers. (Axes and Hammers are two sides of the same weapon class)
Neither Onmyo Magic, nor Spirit, add anything to weapon damage, unless that weapon has the special trait: "Attack increases with Magic" or "Attack increases with Spirit".


Tonfas have B+ for Skill, C+ for Body and D+ for Dexterity. These are the same stats as those of the Kusarigama, but with a different balance.
Odachis have B+ for Strength, C+ for Heart and D+ for Stamina. This is the same layout as that of the Hammer, but with Body replaced by Heart.

 

Scaling Mechanics

Letters like D+ and B+ indicate almost exactly the same ADDITIVE VALUES across different weapons.
This means that if D+ Dexterity scaling adds +100 damage to Bows, then it will add almost exactly +100 damage to other weapons with D+ Dexterity scaling.
Scaling is not affected by the attack rating of a weapon or its damage multiplier, but it is affected by Weapon Level.
For example: 99 Strength adds +226 damage to a level 150 Axe (B+ scaling), but only +76 damage to a level 1 Axe (also B+scaling).
The effect of Weapon Level grows logarithmically, with about 84% of all scaling damage accounted for at at Weapon Level 50.
Past Weapon Level 100, the effect of Weapon Level on scaling is almost completely insignificant.
The difference between D+ scaling and B+ scaling is roughly 1.5 (in other words, B+ adds roughly 50% more damage than a D+)
The difference between D+ scaling and C+ scaling is roughly 1.25 (in other words C+ adds roughly 25% more damage than a D+)
Some weapon classes come in two scaling types. These types alter the balance of the two lowest scaling stats, but never affect the highest scaling stat.
For Swords these types are B+ Heart / C+ Skill / D+ Strength or B+ Heart / C Skill / C Strength.
For Dual Swords these types are B+ Skill / C+ Strength / D+ Heart or B+ Skill / D+ Strength / C+ Heart.
For Axes theses types are B+ Strength / C+ Stamina / D+ Body (Axe) or B+ Strength / D+ Stamina / C+ Body (Hammer).
For Odachis these types are B+ Strength / C+ Heart / D+ Stamina or B+ Strength / C Heart / C Stamina.

The values indicated by "Attack increases with ----Stat----" are a bit lower than the inherent scaling stats of a weapon.
For instance a standard level 150 Kusarigama had +223 damage for 99 Dexterity, while a Kusarigama with "Attack increases with Dexterity (A-)" had +376 damage.
For the buffed Kusarigama, 223 of that damage is accounted for by its B+ inherent scaling, while the A- special scaling added only +153 damage.
The damage scaling for stats on weapons is linear and constant over time (= no diminishing returns). All stats hardcap at 99 (or 200 on Way of the Nioh)
B+ always increases damage by 2.4 x Stat, C+ always increases damage by 2.0 x Stat (At Weapon Level 150).
C always increases damage by 1.8 x Stat, D+ always increases damage by 1.6 x Stat (At Weapon Level 150).
Attack increases with ----Stat---- effects are linear and constant over time. 
Defense increases with ----Stat---- effects are not linear and constant over time.
Change to Attack letters are only approximations of the underlying values (=> A+ is not equal to another A+)
One value of A- Change to Attack was measured to be 0.75 the value of an inherent B+ scaling.
One value C+ Change to Defence was measured to increase Defence with a period of (+2 +2, +2, +2, +3) between 5 and 15 Magic.
The same attribute was measured to increase Defence by 1.65/point in Magic,  by average, between 15 and 35 Magic.
The same attribute was measured to increase Defence by 1.1/point in Magic, by average, between 35 and 55 Magic.
The same attribute was measured to increase Defence by 0.55/point in Magic, by average, between 55 and 99 Magic.
The same attribute was measured to increase Defence by 0.55/point in Magic, by average, between 99 and 200 Magic (Way of the Nioh only)

 

Table Sorted By Weapon

Weapon Skill Strength Body Dexterity Heart Stamina
Kusarigama C+ - D+ B+ - -
Axe - B+ C+/D+ - - D+/C+
Spear C C B+ - -
Sword C+/C D+/C - - B+ -
Dual Swords B+ C+/D+ - - D+/C+ -
Bow C+ - - D+ B+ -
Rifle B+  - C+ D+ - -
Hand Cannon - C+ - D+ - B+
Tonfa B+ - C+ D+ - -
Odachi - B+ - - C+/C D+/C

  

 

 

 

Table Sorted By Stat

Stat Kusarigama Axe Spear Sword Dual Swords Bow Rifle Hand Cannon Tonfa Odachi
Skill C+ - C C+/C B+ C+ B+ - B+ -
Strength - B+ C D+/C C+/D+ - - C+ - B+
Body D+ C+/D+ B+ - -  -  C+ - C+ -
Dexterity B+ - - - - D+ D+ D+ D+ -
Heart - - B+ D+/C+ B+ - - - C+/C
Stamina - D+/C+ - - - - - B+ - D+/C



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