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: 
Yes, there is a difference.

Shutdown will place the kernel in a "hybrid hybernation" mode, so it won't be loaded fully on the next boot, and you will have a faster boot.
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.