The votes on this ticket haven't gone up much recently, and no further comments, so I suspect the need for this comparison information was mostly a problem in the transition period when Bamboo Cloud was being shut down. I plan to close this issue now, with a short word on the differences and our plans for Pipelines.
The fundamental difference is that Bamboo and Pipelines offer distinct deployment options - Bamboo is our on-premises build tool and Pipelines is fully hosted and cloud-only.
In terms of features, Bamboo has a vast existing featureset given its 10+ years of development, and only some of this will ever exist in Pipelines, as a relative newcomer in the very different technology landscape. The features that Pipelines doesn't have (and are likely to implement) are best described by our top-voted feature list. Our mission with Pipelines is to provide great continuous delivery tooling, with a primary focus on teams building modern, containerised web/SaaS applications to deploy in the cloud.
If that sounds like your team, and you want a fully hosted build tool, Pipelines should be the right fit for you. Try it out today, or watch and vote for the features you need from our list, and you'll be notified as we build them out.