Nirnhoned weapon will make just about all your skills hit harder by increasing the weapon/spell damage stat of your weapon. Weapon/spell damage stat is part of the formula that determines actual damage done. But some things do not scale off weapon/spell damage. Set bonuses and enchantments immediately jump to mind. Basically, stuff that scales off quality (white/green/blue/purple/gold) scale off level and quality, not your stats.
Note that higher skill tooltips are not always indicative of better damage. Crit percentage, crit multiplier, penetration, and max resources all play a part in your actual damage done. For example, a skill that does 1000 damage but crits for 2000 damage 50% of the time will be more damage in the long run than a skill that hits for 1100 damage but crits for 2200 damage only 20% of the time. Or, a skill with a 1000 tooltip but gets mitigated to 800 damage because you have no penetration will be worse in long run than a 900 tooltip that hits at 900 damage because you have lots of penetration.
The math gets complicated but do not concentrate completely on weapon damage. Trying to acheive a balance of all the factors that contribute to damage is generally better than going all out on one aspect, even if weapon damage is the most obvious and easiest aspect of damage to understand.