Classic
-Discourages non-combat Stealth use (this being a major feature of the game for many character types)
-Encourages repetition and unnecessary backtracking such as repeating trivial combat respawns
-Encourages tedious actions such as disarming unnecessary traps or unlocking containers of little value purely for XP (and overvalues Lockpicking, Hacking, and Traps skills as a result)
-Gives a consistent drip of XP (this is a nice quality)
Oddity
-Directly encourages and rewards exploration (one of the biggest strengths of the game) as oddity "stashes" are generally behind some obstacle rather than out in the open
-Any character can fully leverage the oddity system, even really bad (according to game mechanics) characters
-Even the most depraved mind hell-bent on gaining XP will naturally progress through the game as they empty areas of oddities
-Oddity acquisition is strongly correlated with progress in the game (breadth and depth), so doing things for the sake of doing them not only uncovers more of this interesting game but also levels your character
The worst aspect of Classic is that it artificially discourages many character builds, but even on builds it's optimal for (like derpy Rifleman) there's also the constant XP nag. "I could do [x] or I could pick up all these traps/go back and do that fight again". Oddity frees the player almost 100% from any concern about XP. Leveling is less of a motivation and more of a by-product of play.
If this was an MMO or something, I could see people wanting to go Classic. But this is an exploration game with a large open world, explorable by a huge array of very different character concepts. The oddity system was literally made for this game.
For those of you struggling with the concept of not being "rewarded" for opening every door or killing every rat: stop playing Farmville on your smartphone and seek professional help.
For the rest which might be skeptical of Oddity: for decades, gamers and game designers have struggled with the simplistic design of XP grinding. This has gone on for so long that people have forgotten that the point of playing games was to experience the gameplay - now we have XP grinding established as a game-within-a-game to the point where in many games that's the only point. A developer has taken it upon themselves to reverse this bastardization, putting an enormous amount of work into creating an XP system that involves neither repetition nor redirection of player energies.
There's no opportunity cost to using the oddity system. If all you want out of a game is to level as fast as possible because of a wicked depravity, the oddity system is the way to go.
This discussion is so insane it's starting to resemble the Grimoire thread. Play on oddity experience. Travel off the path occasionally. Find cool stuff, read cool flavor text. Level whether or not you're trying.