Most of Bamboo's functionality is pre-built. i.e.it doesn’t require plug-ins. It also has built-in deployment products and is integrated with other Atlassian products like Bitbucket and JIRA. In fact, with Bamboo a lot of tasks are available as built-in options. It comes with an extensive menu of tools that you can use to build your application. Jenkins, in comparison, requires that you set everything up – it’s all there, just not built-in.
But one issue, Its not an open source
On the other hand jenkins is a little complicated and difficult to set up as you need to set up all the build-ins and plug-ins but its an open source.
jenkins integration with JIRA and Bitbucket is limited but it's far more customizable and flexible.