Do you think there is any difference between the Quality Centre and Bugzilla?

Yes, of course, there is a difference between both of them.

• Quality Centre is a test management tool, on the other hand, Bugzilla is a defect management tool.
• Quality Centre supports the complete application lifecycle management while Bugzilla is a part of test management.
• Quality Centre also supports the design, execution, traceability and defect management while Bugzilla doesn't.