You can’t link to it.

You can’t really direct people to what you’re talking about.

You can’t make small excerpts to show your readers that it’s actually worth their time to visit.

You can’t bookmark the parts you really like and find them later.

They use a different control structure so they’re harder to use (link aren’t blue and underlined, things generally don’t behave the way you’d expect), showing, in my opinion, that the people who use flash see themselves as more important than their readers/users – i.e. my way is better than what you’ve learned.

It’s not supported on nearly as many platforms, which limits its audience (Flash is one of the few things that still doesn’t work very well on Linux).

It’s a proprietary technology, though that’s a rant for another day.

Search engines can’t search it, which makes it even harder to find it.

Artists and designers are generally the most guilty of using flash for website design, because it does give a very custom feel.  Generally, when I open a site and see that it’s flash, I close it immediately.  There’s a lot of great artists and designers out there that I’d love to share, but won’t because they make it too hard to link to their work.  A lot of great content doesn’t get spread because of a really simple unfortunate design choice.

About the only place where flash is an appropriate design choice: games.  The end.  Nothing that’s not a game should use flash.  Ever.