I go back and forth on this issue. Cartoonish is fun, especially if the game is humorous, in overall approach. The Monkey Island series, and The Next Big Thing, are two examples where the art styles match the theme or "feeling" of the game. Other times, I like a setting that so realistic that feels like a film, or even better, a place I actually am walking through, in person. Some of the Ravenhurst and New York Mystery games hit me this way, as did the recent Portal of Evil game. Where the "eye" of the camera is clearly your own, the sense of immersion is enhanced, versus a game that plays as though you are seated in the audience, watching it unfold on a stage before you.

In each case, though, I want the game to choose one or the other style. Mixing them is jarring, to me, and suggests a game design team without an over-arching idea of what they want their game to be.

I don't rule out one style or another, and enjoy playing both, but feel a game should stick to one or the other throughout, with only a few exceptions that might be necessary to the plot. I can think of a few cases where it worked to be mixing styles (the Puppet Show series is set in a more-or-less normal world but with magical "puppets," for one example).

How do other gamers feel about this? Do you prefer one style over another and do you care if they are mixed into a given game?

IF the story is good, iny ANY game genre, I won't mind the graphics style. Even "mixed "

Cartoonish works for me in the circumstances you describe. Examples from adventure games are the two Wind Monk games and Darkville Castle, whose title I may have just mangled. Examples in casual games are more scarce. I think Royal Honeymoon is one if I've gotten the name right, also Mortimer Becket and the Scruffs.

Photo-realism is a different matter. I'm not a big fan of the visual style in a game like Saga of the Nine Worlds which is photo-realistic (at least for characters and the ship in the opening), saturated Kodachrome although I suppose it's better than fuzzy pinks and purples. I never finished that game. I prefer the looks of Stolen Runes; the less splashy Shiver: Poltergeist and Cursed; and even Death and Betrayal in Romania. Dreary photo-realism, as in the latest MCF, is the worst case scenario, and I fear games are moving in that direction.

That's pretty much where I stand, too.

I agree with you. I was very disappointed when the Grim Tales series went from having "realistic" characters to having goofy cartoons in The White Lady.
For example, check the cartoonish designs of the children at 0:26 and the adults at 6:15 into this video from The White Lady
https://www.youtube.com/watch?v=CGUjJSadgQs

I think the only half decent character design in The White Lady was Richard.

Compare the characters in Grim Tales: The White Lady with those in the previous games.
For example 1:44 into this video from The Vengeance
https://www.youtube.com/watch?v=6HmAwr9lQ4Q

I have to think the cartoonish characters in The White Lady were a cost-cutting measure. The developer must have known it was a mistake because they went back to using more naturalistic character designs for the following game (The Time Traveler).
For example, 14:50 into this video
https://www.youtube.com/watch?v=0SINeiSfxIw

Now if The White Lady were from a new series, and those same cartoonish character designs had been used, I probably wouldn't have minded so much. But a series needs some consistency, and going from detailed graphics to muddy, cartoonish ones isn't likely to impress gamers.

I've seen examples in other games too. Sometimes a game has realistic adults but cutesy, cartoonish children. I guess they don't know how to draw real children.

On a related subject,
There are very few games that can get a pass for using character models that don't fit with the backgrounds -- cartoon characters on a naturalistic background or vice versa. Unless there is some plot related reason, or the game is so obviously stylized that it's clearly intentional, having characters that look like they don't belong in the gameworld looks like either lack of artistic skills or a cost-cutting measure.

Jenny, you've nailed it. Like those in movies and books, game characters should "fit" their environments, unless there is a specific reason for them to be out of place. Otherwise, you are thrown out of the story. I didn't play The White Lady, but what you describe would have put me off, too.