While both are designed to contain files not belonging to the operating system,
/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/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
/usr/localis a local version of
/usrfile system, whereas
/optis place holder for misc stuff.