Bitbucket was a separate product by another company before Atlassian bought them. I would say that that its not just a stripped down version of JIRA but rather an whole other product with an entirely different focus.
The simplest way to summarize is flexibility and customization, and that is primary is different. JIRA's focus is issue tracking and really its strongest powers lie with its workflow and ability to capture required datapoints as well as adapt to different use cases. Bitbucket is a code repository...first and foremost it's job is to function in that manner. Bug tracking and code reviews inside Bitbucket are mainly out of the box with limited functionality.
If all you want to do is to track bugs and some other dev focused bits of work, then Bitbucket will generally meet your needs. At the point where you have <insert strict requirements driven by compliance - e.g. SOX> or different dev teams working in different ways with support expected to function out of the same place, then JIRA is a better system due to its flexibility so you can go from very strict to very flexible.
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