Seriously, why not? It's good stuff. More aliens on the bridge. Worlds not limited by Special Effects budget. And in episode 4, Lt. Uhura takes command!
Put another way, what in ST:TAS is incompatible with the rest of the Canon?
There is no reason that Star Trek, the Animated Series shouldn't be canon. Not a single reason except for the one that DVK jokingly remarks in his comment; because Roddenberry said so. They used the same books, the same references, the same writers.
The writers of the animated series used, essentially, the same writers' guide that was used for the live-action Star Trek: The Original Series. (A copy of the "series bible", as revised for TAS, is held in the science fiction research collection at the Samuel Paley Library, Temple University, Philadelphia.)
The Animated Series was also the first Star Trek to win an Emmy. So conversations around canon are hotly debated with most fans following the party line (that it's not canon) and loving it anyway.
On June 27, 2007, Star Trek's official site incorporated information from The Animated Series into its library section, although it is unclear here if TAS is indeed part of Star Trek canon or not.
But there was far more to the story and ultimately while the issue of ST:TAS being canon has remained hotly debated, facts from the series were paid homage to in all the later series whenever a writer could work it in. So canon or not ST:TAS has managed to affect later versions of the live series.
More DS9 references to the animated series include the episode "Prophet Motive" where the title of healer is resurrected from "Yesteryear" as well. Vulcan's Forge is also mentioned in "Change of Heart", in which Worf wants himself and Jadzia Dax to honeymoon there, as well as in episodes "The Forge", "Awakening" and "Kir'Shara" from Star Trek: Enterprise.
The Star Trek: Enterprise episodes "The Catwalk" and "The Forge" included references to "Yesteryear", the latter featuring a CGI rendition of a wild sehlat. The remastered Original Series episode "Amok Time" featured Shikahr in the background as Spock beams up at the episode's end, and the remastered version of "The Ultimate Computer" replaced the Botany Bay-style Woden with an automated grain carrier from "More Tribbles, More Troubles."
Star Trek (2009) also references "Yesteryear", featuring a nearly identical scene in which a young Spock is confronted by several other Vulcan children, who bully and provoke him for being part human.
Carter Winston, from "The Survivor", has a small but important role late in the 1984 tie-in novel The Final Reflection by John M. Ford. In recent years, references to The Animated Series have also cropped up again in the licensed books and comics. M'Ress and Arex, characters from the animated series, appear in the Star Trek: New Frontier novels by Peter David, in which M'Ress and Arex are transported through time to the 24th Century, and are made officers on board the USS Trident. (David's previous use of these characters, in TOS movie-era comics published by DC Comics, had been prevented by Gene Roddenberry's office.)
A race introduced in the episode "The Jihad", represented by a character named M3 Green, is named the Nasat in the Starfleet Corps of Engineers e-book novellas. These stories feature a regular Nasat character, P8 Blue. The Vulcan city of ShiKahr also appears in many books. Paula Block, then of CBS Consumer Products, was responsible for approving proposals and all completed manuscripts for the licensed media tie-ins, and granted many such uses of TAS material since Roddenberry's passing.
Unfortunately, the issue of ST:TAS lost its canonical status likely due to a personality and licensing conflict that was never resolved successful. Not to worry, in a few decades, no one will remember why ST:TAS wasn't canon and ultimately it won't matter.
All quotes for this were referenced from the Wikipedia reference for Star Trek: the Animated Series.
Canonicity of Star Trek: The Animated Series is disputed, but is generally considered canon except where it conflicts with the live-action series and movies.
So in short, it seems to depend on what you would prefer: Gene Roddenberry's wishes before he died, what the studios seem to be leaning towards, or the middle ground that Memory Alpha has chosen.
( As a side note, while looking for these references, I found several mentions that Roddenberry apparently thought for a while that all of TOS should be de-canonized after TNG started. So that, taken with the 15 years before TAS was de-canonized, makes me think Memory Alpha's stance is probably the most accurate with where it ended up. )
It is canon in every respect that it does not conflict with any of the series.