With a running linux, there are two ways, to mount a harddrive partition correctly. One is using the label (mount -L labelname) or mount using the UUID. The second one is used by debian at first mount (used in /etc/fstab), where you mount the partitions using the UUID. In earlier times, there were only the devices named (/dev/sda1 and so on), but about 2 or 3 years ago, these were automatically renemed to the UUID (some package did this, do not know which one any more).
In practice I am using labels when mounting, when the system had booted up. The only thing, when I noticed a change, is when a new kernel is installed or when doing an update-grub. In these cases, the actual seen device is connected to the other OS (here Windows) and in the grub boot nenu you see, that the device is i.e. no more / dev/sdb1 but /dev/sda1. However, it lokks like only a cosmetic thing (but I am not sure) and could not see any difference. Debian booted, Windows booted. But note, that I am always booting Windows from the grub menu. Hope, this makes things more clear. Hans
