Character Stats affect almost every portion of the game, they include:
Strength affects your damage dealt from physical attacks.
Vitality affects your Physical Defence and supposedly movement speed while unarmed. It also affects your max weight capacity and AC evaluation.
Each class appears to have a different efficiency with vitality in regards to max weight capacity. Fighters receive a max weight capacity increase of about ~0.28 for 1 VIT. Mages receive a max weight capacity increase of about ~ 0.10 for 1 VIT.
Dexterity affects your hard hit rate and increases the low end of your average damage variation.
It also is a major contributor to the damage of Hide Attack, Warzen Blow, Sudden Strike, and Back Attack.
Affects evasion rate as well as variation in damage recieved.
Intelligence affects the damage you deal with magic attacks. These include, but are not limited to, such attacks as: Wind Arrow, Ice Burst, Invest Flame, and Holy Light.
It is also suspected that it has an effect on magic damage you receive.
Piety affects the amount of HP recovered with healing spells. These include, but are not limited to, such abilities as: Heal, Middle Heal, Heal Circle, Middle Heal Circle, Regeneration, and Self Heal.
It is also suspected that Piety affects the magic damage you receive.
Luck has a very observable effect on your revive chance. It is highly suspected that it affects monster drop rates as well. There is currently no evidence that it affects forging results.
(If you perform tests with luck and forging, please link to them in the comments - your data could be useful!)