Flash is made for web delivery and Director is made for CD-ROM (desktop applications).
But it's not as simple as that. Director movie can be delivered through web ("Shockwave"), and Flash can create executable/application for local playback ("Projector").
Flash is far more popular than Director now because web is far more popular than CD-ROM.
The main difference between Director and Flash is that Flash cannot interact with local machine while Director can. Director can create a file, install fonts, or even restart the machine. Flash is not allowed to do any of those things. AIR is invented to fill that gap for Flash but it is not suited for CD-ROM. 3rd party "wrapper" such as Zinc, mProjector, SWFStudio etc. lets you package your Flash movie so that it can behave like Director.
Director is far more versatile than Flash. Director can display almost any media within. All sorts of video formats are supported in Director while Flash can only display FLV (and H.264 since last week). Director can display PDF or even web page. Director can display and interact with Flash, it can be used as a Flash wrapper.
To summarise, there's nothing Director cannot do what Flash can - you can embed Flash after all - but Flash cannot do all of what Director can do. However, if your movie is simple interactive stuff, Flash would be easier to use. Flash is more up to date. Director is still MX2004, although new Director 11 should be out soon.
One other point is that Director needs plug-ins ("Xtras") to perform certain tasks. Director itself is far more expensive than Flash and Xtras can add more to your bill.
Kenneth Kawamoto