Alternate Continuity: The comic series has introduced two key developments that may make it diverge from the cartoon—the Lich is thrown into the sun and the missing chunk of the planet is restored. Also, in issue #11, we find out that in this continuity, Billy is still alive.
An alternate theory is that the events of the comic book take place during the fourth season, before "The Lich".
This states that it is canon, and set during and/or after the show. The blog entry notes that there should only be small divergences due to little finicky last minute additions to the show. Given the two major divergences mentioned above, either this is Blatant Lies or there's a rather depressing amount of Status Quo Is God (if not Foregone Conclusion) for the next few seasons on the show (compared to the comic, anyway).
It is heavily implied that the Lich hasn't come around in a while in the comic. And the missing chunk isn't that big of a deal in the show, it's just kind of there.
The show creators, on the other hand, have explicitly stated that the comics are not canon and are never taken into consideration when making episodes of the show.
Brick Joke: In Issue 27, PB calls Finn and Jake to help out with a ghost problem, except that they're not home because they are the ghosts (unintentionally) causing the problem (don't worry, they get better). So, PB is about to call her back-up choice, which Finn hopes isn't a couple of knock-offs named something like Flynn and Blake. It's actually a team like the Ghostbusters, but guess who shows up at the end of Issue 29, after the storyline is over?
In the third issue, Finn refers to advice given to him by Abraham Lincoln in the past on Mars, which is a hallucination he had in the cartoon's pilot.
The mysterious missing chunk of the planet is explored and filled in.
Issue 7 shows Bubblegum wearing her rock T-shirt from "What was Missing".
Issue 11 references the mess made the last time Finn and Jake tried going into BMO to play video games, in the episode "Guardians of Sunshine".
In Issue 18, the treasure of Hunson Abadeer's branch of the dungeon turns out to be a moldy old pile of french fries, implied to be the same fries of Marceline's that he ate according to "It Came From the Nightosphere" and "Memory of a Memory".
In Issue 21, Princess Bubblegum calls back the events of "Slumber Party Panic" when she says the most the Gumball Guardians will do to trouble-makers is "make you to do their math homework".
Issue 28 features Anti-Ghost Princess, who is Ghost Princess from the same-titled episode back from the dead and busting Ghosts throughout Ooo.
Gilligan Cut: While Finn and Jake explore an abandoned military factory in issue 26, Finn remarks "I can't imagine a single thing our pals could be doing that's more interesting than exploring this crazy old dungeon!" Cue panels of Marceline fighting a dragon, Lumpy Space Princess fighting a robotic version of herself, and Princess Bubblegum... sitting in bed, reading a book.
Issue 25's intro is a shout out to the writer's webcomic Dinosaur Comics. There's even a small caption on the bottom of the page that says: "I am enjoying these... these Dinosaur Comics. HOPEFULLY YOU ARE AS WELL?"
In Issue 26, the machine that turns Finn and Jake into ghosts resembles the machine from Casper that turns ghosts into people.
After the earth is restored, Desert Princess goes missing, and Jake tries to figure out where she went by imitating a princess and saying, "I'm the new princess in town! I'm super nice and everybody likes me!" Cut to the kidnapped Desert Princess, who is bewildered as to why Ice King would kidnap her because she's super nice and everybody likes her.
Stylistic Suck: The Ice King's new fanfiction is about as terrible as the last one.