Dean has black hair. Fangirls have been known to argue over whether Live-Action Dean's hair is brown or blond, but they can probably agree it's not black. For that matter, he now has brown eyes instead of green.
Likewise, Jess's hair color went from blonde to a reddish-brown.
Mary Winchester now has brown hair, which makes Dean look more like their father and Sam look more like their mother, whereas in the live-action show it's the other way around.
Lily now has short brown hair instead of long blonde hair.
All Just a Dream: Jess's reaction to being attacked by a spirit trying to kill her.
Boyish Short Hair: The other characters might look off, but, hey, at least Meg kept her fabulous blonde pixie cut!
Camp Gay: There's a deputy chief in one episode who fits the trope to a T. He lets Sam and Dean out of jail for the reason that they "are so yummy!" Of course, this is thrown into doubt later in the episode, when we discover that he had other motives for freeing them.
Composite Character: The Yellow-Eyed Demon, possibly, given that during his first appearance he is referred to explicitly and exclusively as "the Devil".
Missouri Mosely not only serves the same role she did in the episode she appeared in from the show, but also takes Ellen Harvelle's place during the anime's take on the events of "All Hell Breaks Loose".
Continuity Lock-Out: Given that little to no effort is made to explain what the hell's going on to any new viewers, it seems as though you're expected to have already watched the TV show and just know these things. God forbid someone decides to try Supernatural out through the anime first. Particularly bad in the first episode, which jumps right into a side hunt without laying out the main characters' backgrounds and why there are flaming women on the ceilings.
A Day in the Limelight: Human!Meg and Jake are the centers of their own episodes (Episode 7, "Temptation of the Demon" for Meg and Episode 19, "Loser" for Jake) while John and Lily share "Darkness Calling".
Deal with the Devil: "Temptation of the Demon" follows devout Christian Maria Masters struggling with the choice to make a pact with the yellow-eyed demon to save her dying daughter, Meg. Out of options, she decides to go through with it and becomes possessed by a minion of his as a result, resulting in her attacking Bobby and the Winchesters and dying from the subsequent exorcism. Want to know what puts an even crueler twist on the situation? It is revealed at the end of the episode that her daughter has become possessed as well (casting Meg's "recovery" in a more suspicious light in hindsight) and is killed later on anyway, even after all her mother's desperate attempts to save her.
Eye Scream: In "3 Years Ago", Bloody Mary finds Sam's eyes sufficiently reflective to use as a portal. She doesn't damage them, though. He bleeds from them when she turns her power on him.
Max in episode 11 to his stepmother. Almost.
Gender Bender: The end of "Nightmare" reveals that the yellow-eyed demon has possessed the woman at the drive-thru the Winchesters' orders are at in order to spy on them after the Max incident.
Hide Your Lesbians: Lily was established as having had a girlfriend in the TV series, but her sexuality appears to be either glossed over or outright erased in the anime series, which turns her girlfriend into strictly Lily's roommate/friend.
Ill Girl: Human!Meg has been sick since she was a child. Turns out that she is terminally ill.
Ink-Suit Actor: To varying degrees. Dean and Sam were modeled on their live-action counterparts, though with more emphasis on capturing the character than the actor. Everyone else, no.
Kick the Dog: Max Miller's father is even more of a bastard in this version by virtue of killing Max's dog.
Left Hanging: The show ends where Season 2 of the original series did: the Yellow-Eyed Demon dead, Dean Hell-bound, Sam coming back "different", hundreds of demons let loose on the world...and that's all, folks. It might have been dealt with similarly to how it was in the next few seasons of the TV show, but it's supposed to be an Alternate Continuity, so who knows what would've happened?
Missed Him by That Much: In a flashback episode where Sam is still at college, Dean and their father work a case revolving around Jess, his girlfriend. Rather than disrupt his life again, they decide to just proceed with the case. This involves a lot of Sam barely missing seeing his father or his brother, getting to a ridiculous level when they are one aisle over in the library, not talking quietly, and he doesn't notice a single thing.
The Yellow-Eyed Demon's female host in "Nightmare" and "Darkness Calling" bears such a striking resemblance to Katherine Boecher's Lilith in the original series, right down to the outfit in "The Monster at the End of This Book", that she must have been modeled after her.
In "What Is and What Should Never Be", a photo shows that Dean is a fireman. He wasn't one in the live-action version of that episode, but mentioned wanting to be one in the live-action "Devil's Trap".
Our Vampires Are Different: They even have a difference from the vampires in the original show, namely that they can have kids, who become vampiric around puberty.
Parental Neglect: John Winchester, natch. Also, Maria Masters (Meg's workaholic mom) and Gray Baker (Lily's dad, who abandoned her after witnessing her mother's fiery death).
Pragmatic Adaptation: Most episodes that are taken whole cloth from the live-action show, such as "The Alter Ego" ("Skin") and "Moonlight" ("Heart"), have at least one change to them to help maintain suspense and plot twists.
For the English DVD Release, the original version was used to replace the cover version, and all of the guitar solos are the DVDs menu music. Justified because it was released through the same channels as the main series (Warner Bros, etc), and they had the licence from it that they could use.
Red Oni, Blue Oni: As you can see from the page picture, the official website takes it literally. Their clothes (though not their skin and hair) also tend to show it off in the actual show.
Red Right Hand: Lily's "killing with a touch" power is in her right hand.
Replacement Goldfish: Lial comes to see Sam as a surrogate son that will fill the void her own dead son has left, and wants to keep her with him. Dean and John object to this rather strongly.
Rule of Cool: Due to the nature of adapting a live-action show into an anime, a lot more moments that simply physically can't be done on-screen, either for budget or safety reasons, are added in. For example: Bobby rescuing the Winchesters with a helicopter(!) in "Devil's Trap".
The Stinger: One in each episode. Sometimes irrelevant (a funny moment or the scene where the townsfolk wave goodbye) and sometimes vital to the next episode (like the ends of "Devil's Trap" and "All Hell Breaks Loose, Part One").
Tragic Monster: Madison of episode 10, "Moonlight", is very much this. She was turned into a werewolf and, without even realizing it, murdered her own boss, killed her stalker ex-boyfriend and sadly, in the end, she had to be killed by the Winchester brothers to prevent her from harming anyone else.
Woobie, Destroyer of Worlds: Max Miller of episode 11, "Nightmare", spent his entire childhood being beaten by his father. This left Max introverted and broken. When he finds out he has telekinetic powers he takes vengeance on his father and seeks to do the same to his mother for sitting back and doing nothing to stop his father.