We don't view JIRA and Trello as competitors.
JIRA shines in areas that (a) have workflow and (b) require repeatable processes across a number of people.
Once you have 20+ people on a project, you need repeatable processes.
In cases like bug tracking, project management, customer service, help desks, HR onboarding and hundreds others you need workflow.
Trello shines in areas where you have (a) small teams or (b) require ad-hoc semi-structured data.
In small teams, even if repeatable process would help you, it's not worth the cost of setting up a system - you achieve it by social means.
Trello also has many, many use-cases where you want to start something quick, or personal. In this case it really shines, with near-zero friction to get started.
Scott, CEO Atlassian