I use both actively in a professional context. There are some overlapping features, but in my point of view, they don't serve the same core use cases.
Native Trello is a lot more freeform as far as a Kanban board goes. It's closer to a physical board with post-its stuck to it. You can quickly add rows and cards without being constrained to a set schema as far as the workflow or the fields go. This makes it easier for a user to get started quickly, and less intimidating for personal users.
JIRA, on the other hand, is much more structured. This is to allow a more controlled growth of your project's backlog. The concept of schemas (users, cards, fields, workflows, etc) also allows a company to ensure a certain level of normalization across the projects that evolve within the JIRA instance.
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