Even though Jenkins comes pretty functional and useful out of the box, there's a large plugin ecosystem from which you can choose plugins to integrate into your Jenkins build if you want to extend any of its features.
Unlike some of the simple and hosted alternatives, users need to host and setup Jenkins by themselves. This results in both a high initial setup time as well as time sunk into maintenance over a projects duration.
CircleCI has support only for projects hosted on GitHub so teams that use BitBucket or any other alternative to GitHub are forced to rely on another CI tool or use third-party solutions to be able to integrate CircleCI with BitBucket. One of those solutions may be Cloudpipes.
Bamboo is made by Atlassian, the company that also made and maintains tools such as JIRA, Stash and BitBucket, so it's a given that they would integrate quite nicely. For example, when connecting Bamboo with Stash and JIRA, details like JIRA issues, commits, reviews and approvals follow each r...
Jenkins is a fork from Hudson and as such it inherits most of it's source code. But Jenkins has far more commits and is a lot more active on the development side than Hudson. A lot of plugin developers have also chosen to support Jenkins and develop their product for Jenkins only.