Is there any difference in the way
shutdown + start are handled by a machine?
If it is OS specific, then consider Windows and Fedora.
There is usually no difference between warm and cold boots. When an application calls for a reboot, it usually doesn't care if you restart or shut your computer all the way down and then start. It's probably more BIOS specific than OS specific.
Notable exceptions are hyperthreading and virtualization changes at the BIOS level1. Often, they require a full, unpowered shutdown after committing the changes to take effect.
(1) At least as I've observed with Intel processors.
for the OS there is no difference whatsoever. sometimes the BIOS might do something different if it is power cycled instead of just restarted.
I believe it only changes the way the power/electricity (from the socket or battery) is handled. Everything else till the BIOS (including BIOS) is turned off and then on again in both cases.
The BIOS comes into play BEFORE the OS when powering on, so it is not OS specific.
I am, however not aware of how exactly power is handled, and whether it is handled in the PSU (Power Supply Unit) or on the motherboard.
I am certainly no expert on this, so don't consider this an answer, I'm just posting to say it is not OS specific.