Affinities can modify weapon scaling (what stat influences their damage) or damage type (adding fire damage, magic damage, etc). Weapon Skills are active effects like a unique stab or stomp, and cost FP to activate.
The Whetstone Knife is required to apply Ashes of War — once you’ve obtained it, you can apply Ashes of War to weaponry at any site of grace. Additional Whetblades can be found throughout the game and add additional affinity options when applying an Ash of War.
Note that many Ashes of War have restrictions, and can only be applied to specific types of weapons.