All the answers above seem to be valid, but everyone is forgetting another important point: scaling! Champions in League of Legends have these nifty abilities that often do damage, which is increased by a percentage of a certain stat. For ADCs, this stat is often Bonus AD (as in, AD that only comes from items, runes, masteries and buffs).
Lets look at Graves, first. His Q (buckshot) does damage in a cone, and scales off Bonus AD by 80-144%, in the case of a pickaxe that's an extra 25(.8) - 25(1.44) -> 20 - 36 damage every 10ish seconds. You probably won't get a second chance to back before level 6, so let's tack on his ultimate too: 120-150% bonus AD -> 30 - 37.5 damage. Put these in a combo, and you've got an extra 50 - 73.5 area of effect(!) damage. Hit 2 targets with each (or just one), which is not unreasonable, and all of a sudden you've gotten a lot out damage out of your pickaxe. This damage does get reduced by armor, but your opponent is only auto attacking you for ~50 damage at a time, while you're hitting him for ~40.The auto attack speed steroid that graves gets isn't even factored in here, and that gives even more benefit when you've purchased AD.
For a more common ADC, with no AD scaling, let's look at Tristana: She has a steroid called rapid fire, that at level 1 gives +30% attack speed for 7 seconds. With her q on, she'll get off about 5.9 attacks in the 7 seconds. Without it, she'll get off 4.6 attacks. If we use the same calculations of they attack for 50, you attack for 40, you do 5.9 * 40 = 240, and they do 4.6 * 50 = 230. Chain vest Tristana lost out on 10 damage! Keep in mind, though, that these "attacks per 7 seconds" calculations are done with BASE attack speed, and by the time you have a chain vest / pickaxe, the difference in damage dealt during the trade will continue to swing in the favor of the pickaxing ADC, as they'll have more higher damage attacks.
Armor doesn't work with any of the other stats that ADCs buy: life steal, crit chance (crit damage, in the case of IE), attack speed, Attack Damage works beneficially with all 3!
If the math hasn't convinced you, I offer you this: Do you really want a chain vest, pickaxe and bf sword while the other AD Carry has an Infinity edge? They're melting enemies with their 250% damage crits, while you're... dying slower and doing significantly less damage.