The glib answer is “very little”.
The more constructive answer is that the primary difference between the organization and the community, in my conception of a healthy community, is that the organization is able to provide infrastructure, support, and encouragement that might not have happened otherwise. For example, many communities of like-minded people can’t afford full-time moderation of communications channels, and so even if they want to discuss/work on X, they are vulnerable to attack from trolls, spammers, etc. Or they might be very niche, and so need a full-time organizer to help them do “marketing”/SEO/organizing, without which they might not reach critical mass.
There may be secondary differences as well. The one that comes to mind immediately is preferred forms of participation: e.g., a community might be satisfied with a forum, where an organization might prefer more concrete outputs like blog posts or documentation; a community might be satisfied with unstructured outreach, where an organization might prefer tracking outreach in a CRM.
I’d of course be open to seeing counter-examples, where the goals of community members and the goals of the organization aren’t terribly well-aligned, and yet the community is still healthy/flourishing. But my gut sense is that those are few and far between. (I’ve certainly never participated in one.)