It's important everywhere, for four reasons:
(1) If you write something it's fair to assume that you want to get your point across as effectively as possible. Otherwise, why bother? If you write sloppily then not only are you presenting your point badly, but also you're saying to your readers "my time is more important than yours; you can spend time working out what I'm trying to say because I can't be bothered to fix my errors myself". Right from the start you've crippled your argument by insulting your readers.
(2) For anyone of adequate intelligence it shouldn't take any effort to get the spelling and grammar right 99% of the time. It isn't rocket science. With the exception of those who have genuine medical problems that make expressing themselves in writing difficult, good grammar and spelling should come naturally to every single native speaker of English. It's something foreigners can pick up in a few years of part time study. If you've been speaking English since childhood and writing it for only a couple of years less then you shouldn't be having problems with things like "they're", "there" and "their" or "it's" and "its". They should come to you so naturally that you don't even notice there's a choice.
(3) If you're accustomed to writing badly when you think it doesn't matter you won't notice errors when you think it does matter. Many people ruin job applications because they're so used to ignoring spelling and grammar in their own writing that they never notice when their cover letter looks like it was written by a child. You can't rely on computers: spelling checkers help a little, but they're unreliable and there's not a working grammar checker in existence. I've seen vital business letters sent by corporate directors to important customers where the writing would have disgraced a twelve-year-old. The senders were so used to the "only when it matters" approach that they never noticed the poor impression they were creating.
(4) The very concept that being right is something to be reserved for special occasions is just plain weird. This attitude that being wrong is just as good as being right is far too common these days. Ignorance and sloppiness are not things to take pride in, or to defend.