Is the "Watchmen" TV series a continuation of the movie or the comics?

The series' creator has "explained" this in a letter:

We have no desire to “adapt” the twelve issues Mr. Moore and Mr. Gibbons created thirty years ago. Those issues are sacred ground and they will not be retread nor recreated nor reproduced nor rebooted.

They will, however, be remixed. Because the bass lines in those familiar tracks are just too good and we’d be fools not to sample them. Those original twelve issues are our Old Testament. When the New Testament came along, it did not erase what came before it. Creation. The Garden of Eden. Abraham and Isaac. The Flood. It all happened. And so it will be with Watchmen. The Comedian died. Dan and Laurie fell in love. Ozymandias saved the world and Dr. Manhattan left it just after blowing Rorschach to pieces in the bitter cold of Antarctica.

To be clear. Watchmen is canon.

[...]

But we are not making a “sequel” either. This story will be set in the world its creators painstakingly built… but in the tradition of the work that inspired it, this new story must be original. It has to vibrate with the seismic unpredictability of its own tectonic plates. It must ask new questions and explore the world through a fresh lens. Most importantly, it must be contemporary.

From the December 2019 issue (no. 320) of SFX magazine:

In the years since Lost, Lindelof admits he’d been approached twice about adapting the book, which he says was flattering, but he was too scared to even contemplate it. But after he put The Leftovers to bed, he says the third time really was the charm.

[...]

“I was like, maybe there’s a version of Watchmen, as long as I don’t adapt the original which had already been done in 2009 by Zack Snyder. What if I keep the initial 12 issues canon so that I’m not doing any rebooting? Is there space for kind of a new story? The show could be basically the ultimate intersection between a new story and potentially the continuation of the old story. Because if I wasn’t making Watchmen, I would want to know exactly what happened to some of those characters.”

It is following the 1986 Watchmen comics, not the 2009 movie adaption.

As the word-of-god answer was already given, here is how you can figure this out on your own by just watching the series.

The most glaring difference between the comics and the movie adaption, as pointed out in the question, is the nature of the attack on New York city.

In the comic book, Adrian makes it look like aliens attacked earth by materializing a giant squid in the middle of the city. In the movies, he blames the attack on Doctor Manhattan, basically just blowing up the city and leaving behind some particle traces that look suspiciously Doctor Manhattan-y.

This is relevant as in the very first episode of the HBO series we see

a weather phenomenon in which hundreds of tiny squids rain down on the city of Tulsa, disintegrating just moments after they hit the ground. It is later explained that these kinds of weather phenomena occur on a global scale as an aftermath to the initial "alien attack".

It should be noted that, much in the same fashion as Alan Moore used to radically modernize and often completely change the tone of the comic book series he adapted in the 1980s, HBO's Watchmen takes great liberties with the source material and focuses heavily on themes that are not part of either of the previous adaptions.