Well it depends how you build the site, Flash does not automatically equate to slow, if its built properly, structured properly, designed well etc. then it will be fast.
Anyway, on the whole, as a general rule of thumb, you shouldn't build a whole website in Flash, its not great for SEO or accessibility (though there are solutions to these problems).
To answer your question more generally, Flash has some specific advantages that are relevant to certain types of website and for certain types of content or functionality.
Taking a look at the kind of stuff agencies like BLITZ do, that being highly interactive microsites that are designed to be really engaging and sell a product, Flash still is the best solution out there to do that. HTML 5 comes along way towards it but still misses the mark in terms of what it can do. So Flash has a specific advantage there, but that advantage is heavily decreased by HTML 5.
I still think Flash is best if you need animated content (though this is changing). At the moment, lets say you want an animated banner, in HTML 5; its a somewhat technical thing to achieve as the animation is programmatic. Adobe are updating (I believe Flash Professional) to output to HTML 5, which will give artists the benefit of producing the kinds of animation they would produce in Flash but for HTML 5, which is great. There may be other solutions out there that do this already but the Flash Professional authoring environment is still the best on the market IMO.
There are other areas where Flash has an advantage too, and these are where I think Flash will form a future niche and where it will probably exist for a very long time.
I know we have HTML 5 video which is great however it has its own issues (the codec issue is a massive one no matter how you try to spin it, it's a problem and it needs resolving for HTML 5 video to be really beneficial). Also with Flash Media Server backed streaming, HTML 5 pales as FMS is in a whole different league; it supports HTTP streaming, the same codecs as HTML 5 and it has a whole wealth of other stuff it can do and of course, has awesome DRM and live encoding support; though 3rd party products are needed.
Live Streaming and Chat/Conferencing
HTML 5 and alternative technologies have so much leg way to make to catch Flash / FMS up for live broadcasting and interactivity. Sites like Livesteam, Stickam, BlogTV, UStream etc are going to be on Flash for a very, very long time. The framework and architecture in Flash/FLEX/FMS for these types of site and functionality is just so evolved, mature and fits the purpose so well that it's not going anywhere any time soon, for every achievement made by the competition Adobe make the same and just stays ahead.
I think this is an area where Flash will survive for a really long time.
We also have the topic of Flash (the technology) as a tool to produce smartphone apps, which is a whole different subject area but one where its actually really, really useful.