$$ is a special bash variable (special parameter $ with a preceding expansion mark $) that gets expanded to the pid of the shell.
/proc/self is a real symbolic link to the /proc/ subdirectory of the process that is currently making the call.
When you do ls /proc/$$ the shell expands it to ls /proc/pid-of-bash and that is what you see, the contents of the shell process.
But when you do ls /proc/self you see the contents of the short lived ls process. If you write code which uses /proc/self that code will see its own pid, namely the process making the system call with /proc/self as part of the pathname in one of its arguments.
The $$ is not limited to this usage, you can write echo $$ to see the bash pid; you can use it to kill yourself, etc.