1. If the website visitor is using Firefox and has a not-so-new computer, a flash website or a website with heavy flash animations can take that computer to a crawl.
2. Flash is definitely not good for search engine optimization.
3. Flash is compiled before it is put on the server, therefore it is hard to make changes/updates on a Flash website.
In my opinion Flash can be used -only- when it is a must, there are really good examples on the net like http://home.howstuffworks.com/sewing-machine1.htm
where they use flash animations to show how a sewing machine works. This is the correct usage of flash.