Directory Structure – Difference Between /etc and /usr/local/etc

What? no /bin/ is not a symlink to /usr/bin on any FHS compliant system. Note that there are still popular Unices and Linuxes that ignore this - for example, /bin and /sbin are symlinked to /usr/bin on Arch Linux (the reasoning being that you don't need /bin for rescue/single-user-mode, since you'd just boot a live CD).

/bin

contains commands that may be used by both the system administrator and by users, but which are required when no other filesystems are mounted (e.g. in single user mode). It may also contain commands which are used indirectly by scripts

/usr/bin/

This is the primary directory of executable commands on the system.

essentially, /bin contains executables which are required by the system for emergency repairs, booting, and single user mode. /usr/bin contains any binaries that aren't required.

I will note, that they can be on separate disks/partitions, /bin must be on the same disk as /. /usr/bin can be on another disk - although note that this configuration has been kind of broken for a while (this is why e.g. systemd warns about this configuration on boot).

For full correctness, some unices may ignore FHS, as I believe it is only a Linux Standard, I'm not aware that it has yet been included in SUS, Posix or any other UNIX standard, though it should be IMHO. It is a part of the LSB standard though.

While both are designed to contain files not belonging to the operating system, /opt and /usr/local are not intended to contain the same set of files.

/usr/local is a place to install files built by the administrator, typically by using the make command (e.g., ./configure; make; make install). The idea is to avoid clashes with files that are part of the operating system, which would either be overwritten or overwrite the local ones otherwise (e.g., /usr/bin/foo is part of the OS while /usr/local/bin/foo is a local alternative).

All files under /usr are shareable between OS instances, although this is rarely done with Linux. This is a part where the FHS is slightly self-contradictory, as /usr is defined to be read-only, but /usr/local/bin needs to be read-write for local installation of software to succeed. The SVR4 file system standard, which was the FHS' main source of inspiration, is recommending to avoid /usr/local and use /opt/local instead to overcome this issue.

/usr/local is a legacy from the original BSD. At that time, the source code of /usr/bin OS commands were in /usr/src/bin and /usr/src/usr.bin, while the source of locally developed commands was in /usr/local/src, and their binaries in /usr/local/bin. There was no notion of packaging (outside tarballs).

On the other hand, /opt is a directory for installing unbundled packages (i.e. packages not part of the Operating System distribution, but provided by an independent source), each one in its own subdirectory. They are already built whole packages provided by an independent third party software distributor. Unlike /usr/local stuff, these packages follow the directory conventions (or at least they should). For example, someapp would be installed in /opt/someapp, with one of its command being /opt/someapp/bin/foo, its configuration file would be in /etc/opt/someapp/foo.conf, and its log files in /var/opt/someapp/logs/foo.access.