I imagine you imply that the two DICOM volumes would be from the same imaging modality (i.e. CT, MRI, etc)?
Do you have control over the acquisition parameters like in a standard protocol or this can't be assumed? For example, are the resolution and the sampling along the axial direction the same? What about the voxel size? If these are not fixed, your problem would be much more complicated...
Personally, I would look at the registration literature about the combination of modality/anatomical structure you want to want to compare. The registration error could be used as a measure of similarity between the two acquisitions.
Just to be precise, DICOM is NOT an imaging modality but a standard format to store and exchange medical images of multiple modalities such as radiography, ultrasonography, computed tomography (CT), magnetic resonance imaging (MRI), and radiation therapy.
See the following links for more information:
https://en.wikipedia.org/wiki/DICOM
http://www.cabiatl.com/mricro/dicom/index.html
http://www.dicomstandard.org/current/
What is your specific use case exactly? Are you looking to quantify the differences on 2 acquisitions of the same patient some time apart or trying to find if two acquisitions are indeed of the same anatomic structure (but possibly from different patients)? If so, the meta-data stored into the DICOM file might contain this high level description of the content (like pulmonary X-Ray, ect.) that would be easier to analyze?