One point everyone seems to be forgetting: Originally Flash had a very designer oriented work-flow, so many graphical designers jumped on board and made cool looking stuff, which Internet-users found fun, so they would download Flash, and the community grew.
Java Applets on the other hand were targeted at programmers and enterprises, and that was fine, but boring to the average user. And on-top of that the script kiddies of the time (myself included) made Applets that leaked resources and/or froze the browser, and even other more skilled kiddies used the powers of Applets to create traps, and harmful websites, so most Internet users wouldn't enable Applets.
So in the end it was probably the target developers (graphical designers vs. programmers) of each platform that caused the issues.