When your computer hibernates, the contents of physical memory (RAM) are stored onto the hard drive and the computer is physically turned off. As a result, the system does not consume any more power than if it was shut down (but see exception below).
When you turn your computer back on, early in the boot process, the system will detect that it went into hibernation and will restore from the memory image on the hard drive. A system with legacy BIOS will likely go through POST as if it is starting up from a full shutdown. A system with UEFI will often be aware that the system had hibernated (having been notified so by the operating system) and bypass POST, instead proceeding directly to the bootloader.
Do not confuse sleep with hibernate. Sleep, or standby, mode places the computer into a low-power state to allow you to quickly resume your session. If power is lost, data may be lost as this is equivalent to a hard shutdown.
Newer versions of Windows support hybrid sleep, where the contents of physical memory are written to the hard drive as with hibernation, but the system remains in a low-power state rather than turning off altogether. This allows for quick resuming while preventing data loss should power be lost--the computer will simply load from the memory image stored on the hard drive.
On Windows, the file
hiberfil.sys in the root of the system volume is used to reserve space for the contents of physical memory for hibernation.
Note, however, that if wake-on-LAN (WOL) is enabled, the system will keep the network adapter on to permit the system to resume automatically on a WOL message. Depending on BIOS configuration, the network adapter may or may not be on when the system is fully shut down, and it may be possible to configure WOL in the BIOS setup so that WOL is enabled in hibernate but not when fully shut down. Otherwise, there would be no power difference.
On some systems, and once again depending on BIOS configuration, there may indeed be a nontrivial difference between a full shutdown and hibernate. For example, a computer may keep USB ports powered in hibernation to permit resuming by pressing a key or moving the mouse. In any case, a power failure will not cause a system in hibernation to lose data.