Without knowing the damage formula....
we can only postulate the formulas
There are so many possibilities.

2 possibilities are:

1. Here is my speculated example of the new Formula (Purely Hypothetical, numbers are completely arbitrary and do not directly relate to any in game stats)
*disclaimer: Level May or may not play a part in the formula, however for the sake of this demonstration, lvl difference will be excluded*

This is similar to FFXI, which had a ranged formula.. and a melee formula across all classes

(Damage output + Attack) - (Def) = Delta(Att)
Each value may have a modifier as well as a min/max attached to it

(Str - Vit) = Delta(Str)
Each value may have a modifier as well as a min/max attached to it

the modifiers could even be (and by that I mean probably are) dependent on a dice roll (RNG)

The damage you deal is then made up of some relationship between Delta(Att) and Delta(Str).
For Example: Delta(Str) could define the range of possible damage values and Delta(Att) could determine where in that range the actual attack damage lands
All physical damage would work like this for all classes that can do physical damage

For Lancer, Str and Pie possibly modified by a random number would then apply a bonus to your Auto Attack

2. the second possibility is that each class has a different damage formula Just like each weapon has a different damage formula in FFXII
In this... the 2 stats represent more than just a bonus applied at the end. They are actually part of the initial formula

For Instance your Str and Piety would check against the MOBs Vit....
and your Damage output + attack would check against the MOBs Def

It would make sense seeing that all melee classes have Str as one of their defining attributes