A Cognos cube is based on Cognos 7 technology and is in fact a multidimensional database itself. You need client tools to build and rebuild it not unlike updating a DWH itself (for instance overnight)
DMR is fully Cognos 8 technology and can be viewed as a virtual cube. Each time you adress a DMR model, a query is executed against a relational underlying model. This offers more flexibility (like reporting against 'life' data), but may be less performant.
DMR seems to become quite mature from 8.3/8.4 onwards. Earlier versions are not good enough, especially for sorting data in the GUI.
DMR itself is not a new technique. Business Objects client software relied heavily on similar principles, but Cognos 8 offers a fully web-enabled solution.
A great advantage in DMR is the ability to show attributes nested within dimensional values. This was not possible with cubes.
Ties Blom