Question: In Windows 8, is there a difference between:

*Shutting down* my computer (inside W8) and then instantly *turn it back on* again  

**vs.**  

*Restarting* it from inside Windows 8

Edit: Talking about normal Windows 8 desktop (not RT).  

Edit2: Reason for asking is that shutting down and turning on my computer did not fix something where I needed a reboot, but a *Restart* did. I've heard that when shutting down the kernel is hibernated, but not when doing a restart.

What are the implications of doing one instead of another. When do I *need* to do one instead of the other?

Answer: 
You can turn on/off this under **Power option→choose what the power buttons do→chnge settings that are currently unavailable** and mark/unmark the **Turn on fast startup** option

**What is fast startup?**

> 
> Fast startup is a setting that helps your PC start up faster after shutdown. Windows does this by saving system info to a file upon shutdown. When you start your PC again, Windows uses that system info to resume your PC instead of restarting it.
> 
> 
> 

**What is the difference among shutdown, full shutdown and restart?**

Simple shutdown saves the session in hibernation mode and when you turn back it on it just loads what you want.  

A proper shutdown is doing in Windows 8 by using `shutdown /s` from run dialog or from command line.

While a reboot will followed by a full shutdown with "cold boot".

**When do I need to do one instead of the other?**

Obviously when you need a fast startup then you have choose the shutdown option from charm bar. But if you need a reboot then you have to shut it down using `shutdown /s` command. Also `shutdown /r` will restart the computer with full shutdown. 

You need a full shutdown in case when you have made changes in policies(sometime) or a new hardware.
To do a "normal" shutdown without the need of special command lines (or icons), you can `hold down the Shift key while clicking the shutdown text` over power off menu.  

In response to the main question, yes, default shutdown is realy an ibernation.