"Hibernate" and "Sleep" (also known as "Stand By") are two ways to allow your computer to idle without wasting battery power or having to restart. The differences between the two are minute but important.
Function of Sleep
"Sleep" cuts the power supply to non-critical components and continues to store recent information in RAM, allowing easy access when restarted.
Function of Hibernate
"Hibernate" mode writes the current computer state to a file that is stored on the hard drive, then cuts power completely.
Considerations of Sleep
The benefit of "Sleep" mode is that the computer will restart almost instantly, with the previous RAM storage intact. The consideration is that "Sleep" consumes a minimum amount of power and the computer will have to restart from scratch if the power supply is fully depleted.
Considerations of Hibernate
The main benefit of "Hibernate" is that the current state of the computer is written to the hard drive, allowing it to be restarted faster than from scratch, with all programs running and intact. As well, "Hibernate" consumes no power. The consideration is possible corruption or fragmentation of the saved file if the computer is left to "Hibernate" for an extended period of time.
Windows Vista introduced enhancements to the "Sleep" function on laptop computers. When power is low, sleeping computers will automatically hibernate, which eliminates the necessity of choosing which option is better.