The integrations with the other Atlassian tools (BitBucket, JIRA, and Confluence) work with no effort, right out of the box.
The ability to set fine-grain access control for the deployment phase for each target environment is great. As a public company, being subject to SOX compliance requirements means things such as segregation of duties (no developers deploying to PROD if the application can influence financial reporting). My developers have access to deploy via Bamboo to DEV, but a release engineer can only deploy to PROD.
The configuration of authentication against an LDAP source (such as AD) is a bit clunky compared to the other Atlassian tools. It requires you to configure the user directory source via config files without much direction compared to JIRA and Confluence which you can do it right in the web interface. Even myself being a CLI guy and coder, I would still prefer they were consistent with the other tools.
One big feature that Bamboo is missing is the ability to create a git tag after a successful build against trunk (which is basically a release candidate). Jenkins has this feature out of the box from what I have read. There isn't even a plugin available on the marketplace to do this.
One feature I wish it had is tight integration with Ansible for the deployment phase. Right now I just use a script task for deployment which calls the Ansible playbook to execute the complex, multi-tier deployment with rolling upgrades. Ansible requires SSH authN to the target nodes; if I use the out of the box SSH plugin in Bamboo, I can store the private key within Bamboo, even with a passphrase. If I run Ansible as a script task for the deployment, I can't use the SSH key/passphrase feature of bamboo.
The price is reasonable and well worth it if you already an Atlassian customer using their other tools and want the integration. Jenkins may be enough if you don't care about the integrations.
Consistent builds and automated deployment to all environments (DEV/UAT/PROD). Continuous Deployment in DEV, and push-button deployment in non-DEV by a release engineer (due to segregation of duties requirements). 0 manual steps required other than pushing the button for the deployment.