More specifically how likely it is for me to get published as an independent researcher working as hard as an affiliated researcher? How much difference does it really make?
I would think it depends on the field you're in. I'm in computer security at a major university, and my views will obviously be slanted by what I know.
The first thing you'll need to establish is the resources to which you'll have access as an independent researcher. If you're in computer science or mathematics, this might be everything that you need; if you're in neuroscience or microbiology, not so much. Then remember that 'resources' applies as much to books, articles, and papers as it does to lab resources. In computer science, you might be able to get away with the free resources available; good luck with that in, say, medieval history or medicine.
Second, and more subtly, there's a publication bias issue. In most fields, as far as I know, reviews for conferences and journals aren't double-blind; that is, your reviewers can see your name and (lack of) institutional affiliation when they're reading your paper. Anyone would be biased by that information, and academics are not exception. It's not even a conscious bias, necessarily, but it is going to be there. That means you risk getting rejected out of hand, even if your work is good. By contrast, reviews for the big computer security conferences generally are double-blind, so this isn't such an issue.
It will depend on the field, and the journal in question. Let us assume for the moment that you're capable, without institutional support, of doing perfectly good work, and so the only barrier to publication will be discrimination on the part of editors, reviewers, etc.
There are journals that practice double-blind peer review. One of the best journals in my field does so, so at least from the reviewer perspective, I wouldn't even know you were unaffiliated until after the paper was accepted/rejected. That's relatively common in my field - but there are other fields where its almost unheard of, or where the practice of putting out preprints might make identifying an "anonymous" author trivial.