While there are many places to experience stories and various types of visual art, the way they are presented together in video games is unique. Most significantly perhaps is simply the experience of navigating a fully realised three dimensional world. For instance, I could read lord of the rings or I could watch the movie, both great. But only in shadow of mordor (or other Lotr games) can I experience directly what it's like moment to moment walking around middle earth. And in some ways I can get a better understanding of what the French Revolution might actually have been like by playing assasins creed than by reading history books or attending a performance of les miserables.

A great story driven by my interaction. I enjoy gameplay centric stories like Batman or Zelda or Uncharted, but I also enjoy simple input stories like Heavy Rain, TWD, and Beyond (and we all know the other but let's not mention that by name right now). Now how much interaction varies and it does make me experience the game differently. There's a tradeoff. Gameplay can be satisfying when I overcome a great challenge and yet a simple story driven can be satisfying like a great film is except since I am still inputting and influencing the outcome I am more immersed than with a movie. The little twitches I have to do in heavy Rain or a QTE finisher in GoW makes me feel so much INTO the story than any movie. A great game makes you feel like you are your character no matter its methods.

I enjoy all kinds of games, whether experience(story) or gameplay driven. There's room for both no need to pick sides.

Stories and settings have always been as important as gameplay if not more. People usually fall into extremes when they disscus this issue without realizing that their favirote games they bring up when they try to challenge cinematic games also have the same philosophy in having a story that reflects and complements the mechanics.

Zelda for an example is more of an aesthetic experience than one that challenges your dexterity compared to its 2D counterpart. Majoras Mask certainly has more complex puzzles and Exploration but it's often called the best Zelda game because it had the better writing.

*looks at job*

I care about the money you spend buying them :)

I play video games first and foremost for the gameplay, but I also appreciate good stories in games!

For me, it's a combination of immersion, story, gameplay, thrills and sense of gamer euphoria.

For me this depend the type of game.

Strategy game :  Gameplay > story & visual art 

RPG : I want both on par !

Action/adventure : Story & visual art > gameplay

Fighting : Gameplay > story & Visual art

And sound is really important too, but not in strategy game :)

I want to add, if its not a rpg/strategy game i prefere short good story of 10h, i have hard time to finish long game like gta etc...

I play (I think) to escape life much in the same way an alcoholic or drug addict escapes life by getting slammed and drifting off. For that reason, my favorite games are the ones that are the most engrossing. This could be via gameplay, story, music or whatever. Most of the time, I find that Nintendo's games work best because they are so happy and innocent and completely detached from this thing we call the "grown up" word. They take me back to a place I was at when I was young, for however brief and that brings me joy and peace.

I like mature game as well, but after awhile, they can be too much and I need a break. I can't speak for everyone but, when something is shitty in my life, the last thing I want to do is run around and murder things and be murdered.

Out of sheer boredom ...