(TL;DR at the bottom of the post)

AD and AP play a really interesting role in League of Legends. AD corresponds heavily to physical damage and AP corresponds with magic damage; however, this rule is not hard-and-fast ( Corki deals a lot of magic damage scaling off AD, for example, and Ezreal's Mystic Shot deals physical damage scaling with 40% AP). Also, while physical damage usually comes from physical sources-- hacking blades, crushing blows, etc.-- sometimes these sources deal magic damage ( Malphite is a rock-man who punches people with his fists, and deals mostly magic damage). Most of us accept this and move right along, because to be fair this is a mechanic (like penetration) that is balanced around game balance, not literal or realistic phenomena. But Varus is a bit of a special case, because of his kit: each of Varus' playstyles corresponds to a point in his chronology, and building AD or AP on Varus has deeper implications than merely the type of damage he deals.

Before going into the builds, let's examine Varus' Lore: he grew up in a devout Ionian village, and so it was only with incredible natural talent and discipline he could become a Watcher (unrelated to Freljordian Watchers) over the Pit of Pallas in a high Ionian Temple, a position he accepted with pride. Varus was so enamored with his position that, when faced with the choice to defend his home village from Noxian Invaders and defend his post at the Temple instead, he chose the latter option: duty over family. Turns out that was a pretty bad call, because after coming home to an empty village and his family's broken corpses he flipped out and released the evil within the Pit, the very thing he'd sacrificed his village to contain. Real smart decision, there, buddy.

Anyway, the important bit here is that Varus' "fighting style" has effectively changed over time. Pre-corruption, Varus protected his temple from scores of Noxian Invaders, and post-corruption he mowed down their ranks in a fury:
Though what it fired wasn't an arrow, it behaved as one; it ruined as one, traveled through the Noxian insignia, out of one gasping soldier and into another, exploding with six thirsting tongues that gripped and snared. With every death Varus shot faster, until by the end there was no sense that each shot was discrete. The Prey simply ran until it fell. It was elegant horror.

Now, this "rapid firing" isn't consistent with Mid Varus' poke playstyle. It is, however, totally in line with ADC Varus' playstyle, because with Runaan's Hurricane and an auto-attack focused build, Varus can attack multiple targets in flurry similar to that described here. I'd compare Mid Varus more closely with the hunter he would become as time passed, no longer reaping hordes of soldiers but ambushing select targets to gather information about top Noxian agents:
Time passed quickly. There were more hunts, and more blood; mercies were promised, but withheld. Darkness swirled with the scent of wet leaves, oddly sweet, the broken bodies of men and women rendered wholly abstract in the course of their annihilation. Varus knelt over a sundered cart to seize a communique, pierced through by one of his sharp tendrils; it revealed the names of yet a few more Noxian dogs, responsible for issuing the order...

So the separation between the playstyles isn't solely a side effect of League of Legends itemization, game mechanics, and balance: it's also tied closely to Varus' actual lore. There still exists one style not covered by this interpretation: AP Varus, a version of Varus that doesn't use Piercing Arrow or auto-attacks for damage, instead relying on proccing Blighted Quiver and Chain of Corruption. Maybe you can see where I'm going with this by now: AP Varus is the end result of Blight consuming Varus, after either his vengeance is achieved or sufficient time has passed. When you think about it, it makes sense: building no AD, his power cannot possibly derive from his own strength as an archer; as the Blight has consumed him, he struggles even to draw the bowstring. Most of his damage scales with the target's max HP, which fits in with the theme of consuming the target on another level, damaging their bodies on a level that cannot be defended against. The fact that his Chain of Corruption, a physical manifestation of Blight, scales 1:1 with AP just makes this perfect.


TL;DR: To summarize, ADC, AD Mid and AP Varus actually correspond with different points on Varus' timeline, and his style of killing matches his build paths. This is one more reason I love playing Varus: as I auto-attack with Runaan's I can't help to imagine Varus frantically slaughtering the terrified Noxian Soldiers in their retreat, and when I explode someone for 50% of their max HP as AP Varus I imagine them being literally consumed by Blight. Mid Varus' approach is more methodical and foreboding, and I'm less interested in that myself, but I think it no less appealing gameplay-wise.

EDIT: Linked quoted text to the source. If you haven't read it yet, give it a read-through: it's written by Tycho (Jerry Holkins) of Penny Arcade.