Basically it is the same. The only difference is that C+R can produce some very small fragments so if this is paired-end sequencing then some people tune aligners such as bowtie2 a bit, see e.g. Cut & run - bowtie2 dovetail option but I am not sure whether this is truely necessary. For peak calling there a some C+R-specific callers such as SEACR that claim to perform better with very high quality data which have extremely low levels of noise, hence might overinterpret small peaks. Anecdotically I heard that many use macs2 though as often C+R quality is just the same as ChIP-seq, and the few C+R datasets I had a look at worked quite ok with macs2. Once you have your peaks it is the same as in any other (differential) analysis if you ask me. That all is just my "thinking aloud" so take it with a grain of salt. I mean after all it is pulldown of protein-bound DNA and then sequencing, there is no fundamental difference between the two assays other than that ChIP does crosslinking and sonication on extracted chromatin while C+R uses MNase-based cutting without crosslinking on permeabilzed cells. It is still an antibody-based assay so quality widely varies and depends on epitope uniqueness, abundance and the daily mood of the antibody god. In my limited experience C+R, while a lot more convenient on the lab side as it is super fast and required few cells, is not a guarantee for a good quality dataset (not that you were asking about that but I felt like writing about it :) )
