When you restart your computer after a normal shutdown you only load the operating system and the programs in your startup folder are started fresh.

When you restart from hibernation, all the programs you had running are still running exactly as you left them. So, for example, if you are writing a document in word and you are typing on paragraph 3 of page 28, then you hibernate; when you restart word will be open with the cursor on paragraph 3 of page 28, even if you didn't save the document.