Writing maturity means a few things. It determines how you deal with critique, how you give critique, how you deal with other writers' successes and failures, and most importantly, the view you take on your own writing.
The normal kind of maturity that everyone has is normally linked to writing maturity, but not always. I've seen people who are the most mature people in the world--until they get critique on their writing, and they turn into a petulant toddler. I've also seen people who aren't the most mature person ever, but take critique like a champ and generally have a critical but understanding eye for their own work. And I've also met people who don't have either. (I tend to avoid them.)
From what I've seen, the people who have a high level of writing maturity are the ones that progress that fastest in terms of their writing skill. They're open to all kinds of critique, they're generous with their gratitude, and they're down-to-earth about their stories. (At least on the inside; it's okay to have a giant ego about your writing sometimes--it's called self-motivation.) Because of this, they're more flexible. They grow with each piece of critique, and they realize that they aren't the Big Kahuna of Writing Knowledge. They're humble, but not stupid. They know what's good for their story and what's not.
One person I know who has a high level of writing maturity is my CP, Leigh Ann. I got to read both her first and second MSs, and I was floored by her increase in skill between them. Part of that increase was because she's a hard worker. The other part came from her openness to critique and her discerning eye when it comes to what her stories do and don't need. She is always very thankful when receiving critique and generous when doling it out. (Seriously -- just look at her CP Fanbase. And don't contradict me, L. We are basically your fanbase.)
People with little or no writing maturity stagnate. They can't rise to their full potential because they aren't open to the fact that someone else's knowledge and expertise can help them. When you're at this point, the most crucial thing you can do is mature. Mature with critique, mature with your writing.
Basically, what I'm saying is,
DON'T BE A DOUCHEBAG.