HTML website: The Fruit Source
Flash website: Viki Music
HTML website with some Flash elements: Rune Steiro

Flash websites are great for:
• gaming
• entertainment purposes (music bands or movies sites)
• obscure concept sites

Some of the disadvantages of a Flash website:
• they take longer to develop and, therefore, are more expensive
• they are harder to update
• they do not rank well with Search Engines
• users can’t bookmark specific pages or copy text
• they take a longer time to load

Most users prefer websites that get to the point straight away. These sites are simple and eye-catching and they load fast. No one will wait for your site to load and even the coolest animations are annoying if the user is forced to watch them repeatedly. Brands like Nike can have a full Flash website because they are a household name already and don’t care much about their Google rank.
Should you build an all Flash site? Probably not. I would rather recommend using Flash features within a regular html/css site.

More in-depth analysis : The Serious Drawbacks to Using FLASH for Web Design