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: 
Straight from Microsoft Blog:

> 
> **Here’s the key difference for Windows 8**: as in Windows 7, we close the
> user sessions, *but instead of closing the kernel session, we hibernate
> it*. Compared to a full hibernate, which includes a lot of memory pages
> in use by apps, session 0 hibernation data is much smaller, which
> takes substantially less time to write to disk. If you’re not familiar
> with hibernation, we’re effectively saving the system state and memory
> contents to a file on disk (hiberfil.sys) and then reading that back
> in on resume and restoring contents back to memory. Using this
> technique with boot gives us a significant advantage for boot times,
> since reading the hiberfile in and reinitializing drivers is much
> faster on most systems (30-70% faster on most systems we’ve tested).
> 
> 
> **Use of multiple cores**: It’s faster because resuming the hibernated system session is
> comparatively less work than doing a full system initialization, but
> it’s also faster because we added a new multi-phase resume capability,
> which is able to use all of the cores in a multi-core system in
> parallel, to split the work of reading from the hiberfile and
> decompressing the contents. For those of you who prefer hibernating,
> this also results in faster resumes from hibernate as well.
> 
> 
> **Shutdown+TurnOn≡ColdBoot**: Another important thing to note about Windows 8’s fast startup mode is
> that, while we don’t do a full “Plug & Play” enumeration of all
> drivers, we still do initialize drivers in this mode. Those of you who
> like to cold boot in order to “freshen up” drivers and devices will be
> glad to know that is still effective in this new mode, even if not an
> identical process to a cold boot.
> 
> 
> 

And this point is worth mentioning regarding your Edit2 where shutdown+trunOn didn't fix your problem, but Restart did:

> 
> **When to use Restart specifically**: Of course, there are times where you may want to perform a complete
> shutdown – for example, if you’re opening the system to add or change
> some hardware. We have an option in the UI to revert back to the
> Windows 7 shutdown/cold boot behavior, or since that’s likely a fairly
> infrequent thing, you can use the new /full switch on shutdown.exe.
> From a cmd prompt, run: shutdown /s /full / t 0 to invoke an
> immediate full shutdown. Also, *choosing Restart from the UI will do a
> full shutdown, followed by a cold boot*.
> 
> 
> 

More extended info, visit: [Delivering fast boot times in Windows 8](http://blogs.msdn.com/b/b8/archive/2011/09/08/delivering-fast-boot-times-in-windows-8.aspx)
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.