Stand by does several things to save power without actually shutting
down the machine. They may include turning off the disks and turning
off the monitor.
Hibernation involves writing the current contents of RAM and video
memory to the disk, then shutting the machine down. The next time you
boot up, the boot-up code detects the hibernation file on the disk,
and loads that to restore the state prior to shutdown.
Going into and out of standby is a lot faster compared to hibernation.
If you lose standby power from the power supply while in standby, the
computer will no longer be in standby. A computer in hibernation
remains in hibernation even if standby power is lost, so hibernation
is safer.