Repair permissions: device vs volume


In Disk Utility, is there a difference between repairing permissions with the device selected vs a volume on that device?


No, no difference.
The choice will often be greyed out, if Repair Permissions is not relevant.
For example, Repair Disk Permissions can't be run on a volume that does not have an OS X system (not an OS X boot volume)
I have an external with 10 partitions. 4 are bootable. Repair Permissions is only available when one of those 4 is selected, and also not available when selecting the device.