Difference between shutting down and turning on vs. restart in Windows 8?

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?

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

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.