Sleep
The computer enters a low power mode that keeps everything in RAM but "pauses" execution of programs. Various things can wake the computer up from this state such as just about any interaction from hardware depending on settings.
Hybrid Sleep
The computer stores the contents of memory to the hard drive to prevent data loss but also leaves the contents in memory to allow resuming faster.
Hibernate
The computer stores the contents of memory to the hard drive and powers off. The only way to wake up from this would be through the computer's hardware. For instance, you can look in the BIOS/UEFI to see what can wake the computer up. Newer computers have systems such as Intel Active Management Technology that allows the computer to be woken up or put into low power states where updates and maintenance can be done. Wake-on-LAN, wake on keyboard/mouse, wake on network activity are all features that can be found in some BIOS/UEFIs that will wake a computer in hibernate.
If the computer is set to hibernate, Windows is inactive and cannot be interacted with++. With that in mind, Windows won't be able to sleep since it's not running. If Windows is sleeping, it's still powered on but in a "paused" state. Various things will bring the computer out of this state. Since Windows is still running, it can switch from this state into hibernation. For instance, you can configure Windows to bring itself out of sleep to perform Windows Updates.
I believe Windows uses the same information to determine when the computer is idle so sleep and hibernate timers start at the same time. I can't find a source directly confirming this (see links 2 and 3).
++ excluding out-of-band management such as some systems that Intel has among others
Sources:
http://www.thewindowsclub.com/difference-between-sleep-hybrid-sleep-and-hibernation-in-windows-7
(System Wake-up Events) https://msdn.microsoft.com/en-us/library/windows/desktop/aa373235(v=vs.85).aspx
(System sleep criteria) https://msdn.microsoft.com/en-us/library/windows/desktop/aa373233(v=vs.85).aspx
(System sleeping states) https://msdn.microsoft.com/en-us/library/windows/hardware/ff564575(v=vs.85).aspx
(System power states) https://msdn.microsoft.com/en-us/library/windows/desktop/aa373229(v=vs.85).aspx