I have extensive experience with both. A concise reply is that Job DSL has existed for much longer and was Netflix's open source solution for "coding" Jenkins. It allowed you to introduce logic and variables into scripting your Jenkins jobs and typically one would use these jobs to form some sort of "pipeline" for a particular project. This plugin received quite a bit of traction as a common way to enable job templating and scripting.
Jenkins Pipeline (2.0) is a new incarnation of a Jenkins job that is entirely based on a DSL and attempts to eliminate the need to stitch together multiple jobs to fill a single pipeline which was by far the most common use of Job DSL. Originally, with Pipeline DSL not offering many of the features that Job DSL did, and as mentioned above Job DSL would allow you to create Pipeline jobs, they could be used together to define a pipeline.
Today, IMO there is little reason to use Job DSL because Pipeline is the Jenkins-supported mechanism for scripting Jenkins pipelines and it has met or surpassed much of the functionality of Job DSL. New plugins are being developed natively for Pipeline, and those that don't are being encouraged by Jenkins developers to integrate with Pipeline. And Pipeline has several advantages:
Finally, Jenkins Pipeline is by far the most prevalent feature of Jenkins right now. Check out the Jenkins World 2016 agenda and you'll see approx. 50% of the sessions involve pipeline. None for Job DSL.