I'm looking for artsy / experimental video games with an emphasis on atmosphere or story.

-The shenmue series is a great adventure game that emphasis atmosphere and story.
- Oni despite being an action title is big on story. It plays out like a anime movie almost.
-Chulip is a weird game about getting hearts so your character can have the courage to tell his love that he love her.
-Chulip is a platformer that about being a cleaning robot that helps bring the family together
-Killer 7 is a dark title that has stylized graphics and a them of politics.
- Any game by the developer spike and it's sibling companies
-The SOS series is an interesting chose your own adventure disaster survival game.
-Pathologic and the void are strange experimental titles that are far from traditional. Both are fps but they are about atmosphere.
-Shadows of destiny is a game about changing your fate. There are multiple endings depending on what you do in the past.
There are plenty of games out there, you just have to find them.

I was going to recommend this one as well. It's a free download. Here's the link:
https://typo3.digipen.edu/index.php?id=986

How bout some Yume Nikki?

Try The Void. Definitely the weirdest gameplay I've ever seen. Very atmospheric.

"Hamlet, Or the last game without MMORPG features, shaders and product placement"
Or the short name: mif2000's Hamlet
http://www.alawar.com/game/hamlet/

LOL That's an awesome game. I know the guy that made it. :) I'll tell him he got mentioned.

Seriously check out The Void, it fits the bill perfectly.
I am currently playing it through with cheats because the atmosphere and art hold the game up on their own. After three tries I am tired of playing the beginning over and failing later. I suck at it but I am in awe of it.
Also, coolest cover art ever.

Agree, The Void.
Also, if you have Steam and a Half Life 2 game, you can download these free:
Dear Esther
Korsakovia
XIII is interesting in that it did cel shading in games before Borderlands and Prince of Persia
American McGee's Alice is an interesting dark spin on the classic story
Saira is a great platformer

Digital has already been mentioned, but I played it Sunday night and really enjoyed it. Some of Emilia's writing is rather squishy (she's a novice poet, after all), but the game is pretty engaging, both for its internal story and as an examination of how we relate and communicate online. Anyone who's ever found a friend online will be able to relate to Digital.