It's almost never necessary to restart after installing. It almost never was except in the case of hardware drivers. I believe now even modern OSs (like Win 10) doesn't even need a restart when that happens now.
There are some RARE cases where restarting is necessary. These "restart before running" messages were common with the install wizards of the day just as a "cover your bases" step. It was once enforced (possibly practiced) that changes to the registry required a reboot to take effect. However, adding branches to the registry as in the case of FFVII and FFVIII won't affect anything that would require a restart.