Do you care about your Pokemon's abilities or natures?

It honestly depends. If I am playing through the regular part of the game then I will go for the certain ability I want but I am less inclined to search for nature. Yet if I am preparing a team for battling I will make sure I get right nature and ability.

Not really, but if they do ever have a good one it's a bonus rather than a feature I'd search for.

When I'm playing through the game's storyline, I don't really care... I take any nature so long as it doesn't lower the Pokemon's best stat. For example, I probably wouldn't train a Tepig with a Calm nature (as its attack is lowered).

Once I finish the storyline, yes, I do care a lot. I start breeding a brand new team to get a fitting nature and characteristic and then EV train.

I remembered reacting to my Jellicent having an Adamant nature, which is terrible as they are mainly Special-based attackers. I went "no, I can't use it cos it has a bad nature" so I decided to go around looking for Modest Frillish and to help me, I caught lots of Elgyems and I took one with a Modest nature and Synchronize ability to help me find the perfect Frillish. I eventually caught a Modest Frillish and evolved her into Jellicent.

Yes, I caught lots and lots of Elgyems and there is a reason why I have them all. I can take the ones with Synchronize to help me find particular natures. I guess they do come in handy after all.

So, if you're playing Black or White and you're trying to find a particular nature I recommend going to Celestial Tower and catch some Elgyems and put the ones with Synchronize with the nature you're looking for (make sure it has Synchronize) at the front of the party and eventually you'll find the perfect Pokemon.

Also, if you catch lots of the same species they will always have differences in stats even if they are the same nature. So I also compare them.