Damage obtained from chest/pants/set bonuses, and other buffs like the flat damage value form ribs, make a large diference. As already stated,
Main Hand Damage Per Second = character screen damage / current attack speed
This means by incraesing your damage, or "improving" your attack speed both result in more main hand dps, and thus more damage output from your skills. However, attack speed will NOT increase the amount of damage inflicted from a single white attack (this amount stays the same when attack speed changes, but you attack more frequently instead).
An easy way to see this effect is to equip a level 1 weapon, and hit a monster. Then remove your chest or pants that give bonus damage, and hit the same monster. You will notice a substantial difference. With more % damage buffs, such as the 12% bonus from ribs, or 20% bonus from champion/priest buff Suicide Advance, the bonus damage from non-weapon sources grows substantially.
An important thing to note, is that bonus damage grows twice with each "% damage increase" buff. For example, let's say your chest piece gives you 5% bonus damage when equipped, vs not having it equipped (10 000 vs 10 500). With tambourine buff (15% damage), the difference with chest equipped might be 11 500 vs 12 161. This is a 5.75% increase in damage with/without chest, compared to the previous 5%. If you wish to fully understand how character screen damage is calculated, and how bonus damage factors into this, see the first link Cike posted. With enough buffs (ribs, c/p wd/s k/p ibw tambourine) this difference can grow from 5% to 9% of your weapon damage. (the 5% is just hypothetical, the 9% increase is a value calculated based on the hypothetical 5% base ; real values will be different).
tl;dr bonus damage is super good, and gets EVEN BETTER with % damage increase buffs.
ps. Holy crap, I totally forgot about those adf spreadsheets....So much work put into those lol. Turns out, they needed some fixing. They're still a mess, and aren't meant to be a guide by any means.
This post has been edited 3 times, last edit by "BlankMinded" (Mar 30th 2016, 2:06am)