/proc/kmsg provides a root-only, read-only, consuming view of the kernel log buffer. It’s equivalent to calling syslog(2) with the SYSLOG_ACTION_READ action. As mentioned in the proc manpage,
A process must have superuser privileges to read this file, and only one process should read this
file. This file should not be read if a syslog process is
running which uses the syslog(2) system call facility to log
kernel messages.
/dev/kmsg provides access to the same kernel log buffer, but in an easier-to-use fashion. Reads are tracked per open, so multiple processes can read in parallel, and entries aren’t removed from the buffer as they are read. /dev/kmsg also provides write access to the log buffer, so it can be used to add entries to the log buffer. See the /dev/kmsg documentation for details.
As for why both are present, and why one is in /proc (albeit not process-related) and one in dev, /proc/kmsg is an old convenience “export” of kernel internals, and /dev/kmsg is a more recent addition, designed as a usable interface to the log buffer.