Rolls basically come in three flavors: “rolls” (e.g. attack, damage), “checks” (e.g. ability score checks, skill checks), and “throws” (i.e. saving throws). A circlet of persuasion only applies to the “checks,” and not to “rolls” or “throws.” And it certainly does not apply to anything based on Charisma that doesn’t involve rolling a die, like for example a sorcerer’s bonus spells or save DCs. Basically, you need to see the word “check” somewhere if you want it to apply.
But it does apply to all checks involving Charisma, rather than only ability score checks. The wording for that would be a “bonus to Charisma checks” rather than a “bonus to Charisma-based checks.” So yes, it does apply to skills and some other things.
Major examples:
Ability score checks, e.g. the Charisma check made during planar binding
Skill checks, e.g. Perform or Use Magic Device
Concentration checks, e.g. for a sorcerer
Major examples that are not checks:
Non-dice-based things, like bonus spells for a sorcerer or AC for an oracle with the nature’s whisper revelation
Attack rolls, e.g. with smite
- Note that performing a combat maneuver involves rolling “an attack roll and add your CMB in place of your normal attack bonus,” so it
Damage rolls, e.g. with Desna’s Divine Fighting Technique and a starknife
Saving throws, e.g. with divine grace