Hibernation means that when you go into SLEEP or STANDBY mode the data in you system memory (i.e. DDR3 sticks) is copied over to your system HDD/SSD.
Thus,
Advantage: comes out of STANDBY quicker
Disadvantage:
- slower entering sleep or standby (which only matters for SLEEP if you sit there waiting to ensure it did so)
- SSD usage (wears SSD out a bit which shouldn't matter to most people)
- some capacity used (if you have a small SSD it may matter)
Some people run out of SSD space then disable Hibernation to help clean up a little space. At that point I'd be buying a new SSD myself if cleanup doesn't help enough but it's a good solution for some people.
Other:
If you shut down the system completely Hibernate isn't used. So it won't affect shutdown or startup in this case. It is only used in SLEEP or STANDBY modes.
SLEEP - power off
STANDBY - power on (Hibernation is used in case of power loss)
If you come out of STANDBY then the Hibernation file is not used since the data is still in the system memory.