Attack damage perks only increases the damage per hit for the king’s melee attack.
Depending on the damage type(s) on your items, that damage may be more or less effective against troops and towers (e.g. most/all towers are weak to blunt damage, but not to the regular damage).
Spell perks generally increase the spell’s “power”, which is usually its damage - although there are some exceptions to that, e.g. shield /heal spells grant more shield / healing, obviously, and for black magic you get higher morale counts that you can copy.
The increase is a % value proportional to the numeric perk value. By comparing a spell’s stats when viewing it once from the wizard tower (regular power) and once from the attack preparation screen (boosted power with item perks), you can see the difference in values. Multiple perks of the same type will stack.