Blackfire is a profiler. It measures code's resources consumption at run-time. It enables users to write custom performance tests, as well as receive performance optimization recommendations. You can read more about it here: https://blackfire.io/docs/introduction.
Overall, Blackfire helps to ensure that the back-end part of an application runs fast for an optimal end-user experience.
Overall, Insight helps developers have cleaner, better structured, and safer code.
Complimentary and not Competitive
While there is a bit of overlap between the two services (both examine the code and can provide recommendations for "best practices"), they work very differently. Insight looks at the code "at rest" where Blackfire looks at the code while it is running.
Most projects benefit from having both a static analysis tool and a profiler in their toolbox.