I don't know whether there's a large correlation, but I believe it would mostly be in the severity of outcomes. Depression isn't "because life is hard", but life being hard makes dealing with depression harder than life being easy.

I know people who are very wealthy who struggle with depression, I know people who are very wealthy who don't even really understand the term because they've never experienced anything remotely close. And I know both types who are not wealthy at all. From my personal experience, money doesn't matter in that regard. But it's definitely better to struggle with depression without the added stress of keeping up with the bills, and when you've gotten through it, you're having a good life if you've been rich before, and you're going to have to pick up the pieces and try to glue them back together if you haven't.