What do you care more about in a character?

In my opinion, the way the character looks is more important because that's what will capture someone's attention at first sight. If there's a character that looks extremely boring, even if they have the most amazing personality and story, it won't get the attention it needs in order to find out about said personality and story. You can always add more stuff to your character's backstory later on, but once you make an extremely boring looking character, you can't make it look better unless you completely redesign it, which doesn't really count as the same character design anymore. So, when you make a character, make sure it looks good first, and then you can worry about story later.

Obviously, the ideal character is the one that looks cool and has a good backstory. But if you had to choose just one, which one would you choose?

Thomas was alone has a great set of characters that look all like squares, but are all uniques and appealing thanks to their personalities. Colors and shape are just here to differentiate them. We can say the same for comics like xkcd or even Peanuts, where (almost) only accessories, clothes and hairs help to recognize characters that are defined by their personalities.
The Scythian from Sword and Sworcery EP is also a great example of character writing: the character is so pixelated that we can't really see much about her, and she literally has no back-story… But it's her actions in the game that build her character, and allows to see who she is! It's a character defined not by what she did, but what she does.
And anyway, in literature, if you think about it, there's not really strong characters design. Sure, the author can write a description of its characters, but it's not uncommon to see some characters appearances let to the reader's imagination. Some characters are awesome because of who they are instead of how they look like.

Although, the look comes as primordial when you consider multiplicity. Characters have to be different to each other, not only for a practical reason (I hate those movies where every actors and actress look exactly like the same model of perfection) (I'M LOOKING AT YOU PACIFIC RIM), but also for the universe's depth. A world with only one kind of people is boring, not credible, and in the worst case exclusive. That's where diversity comes in handy! \o/ Characters must have main traits that define them in the story, but also secondary one that aren't pertinent, but make them more complex, unique, and overall likable. Even if it's not the definition of a character, it's always good, as an author, to gives some supplementary aspects, as long as it's not in contradiction with what they are.
Generally, I try to write arbitrarily some of my characters into women, even if I imagined them as men. Since their main aspect wasn't masculinity (for most), it works most of the time, and give a set of characters richer than what I attended. And it's even better when I give a character Russian origins, another one short legs, or when I add some different colors, accents, age, weight, physical handicaps… It's a way to make them identifiable, and to create a universe only made of people like myself.

So basically, that's the order I chose to create characters: first, the personality, its story, the main definition of who they are.
Then, I add to them aspects, traits, optional details, that allow to make them recognizable in the cast. Those aspects are made in function of the others: I don't want two characters to look alike, so I make them different. But who they are they are is for the most part independent of how they look like.
Well, more exactly: who they were when I first write them. It's an evidence that adding those traits have an influence on the character itself. But it doesn't change the core, and make even in the end a new character built from several ideas. And that's something I find very interesting: discovering my own characters as I assemble them!

Well yeah, I'd pick personality over looks, especially since literature/movies/games/etc open up a very different way of connecting and observing people than we would in real life: Whether they admit it or not, you don't need a social experiment to confirm that in real life, most people will show more attention, interest and respect for when someone fits their image of appealing looks.

This isn't really the case with a written story, where you can work in characters that have low self-esteem, treat others poorly, and/or be physically unappealing and still allow the reader to become involved in their world (Ex: Harry Potter's relationship with Hagrid, a character that others judge poorly due his appearance and being accused of fitting a stereotype that claims he's dangerous and aggressive). TL;DR, a "cool looking" person in a story can be a lot of different things, whereas most people in real life will simply stick with their preferred "kind" of person.

And also because, well, a lot of writers don't go much deeper than appearances. A lot of movies bring in the girl character that only exists to look pretty and they never receive any character development, or they bring in characters that just fit a stereotype, and thus the appearance connected to that stereotype is being relied on to make people get an idea of who they are.

And hey, a lot of books aimed at teens are all about average looking, insecure teenagers. They don't look cool, but that just helps them connect to angst-ridden, "not in the popular pretty club" teenagers. Even though the character may be completely uninteresting design wise, they are still cool, or at least relatable in the mind of the reader.

I'd choose looks over personality. These aren't real beings, so I can afford to be shallow here.

I'd pick a silent but visually interesting character over a pink fluffy thing that engages well with the player/the storyline hands down. Personality is important too, but the character and the game has to look striking and cool first and foremost, as the OP suggested.

Usually, artists draw character in a way that makes their looks reflect their personalities and backstories. So if the two aren't in sync, it's usually just poor artistry or poor writing.

In my opinion, they're almost equally important. But, the thing I respect or disrespect most in character is their character development or lack thereof.

Character's looks. Take a random character as an example : Tily. That's just impossible to like a such monster.

Personality and story of course. I don't care if character is designed good if he is asshole to others and if his backstory is lame.

The story.

The background colors.