Younger readers may not fully appreciate how much the internet has changed over the years. When I started using the internet in the nineties it was done with a device called a modem which plugged into your phone line meaning that when you were connecting to the internet you were making a phone call to your internet service provider. The modem I originally had was a 28000 baud modem which means it was able to send and receive 28000 bits or 0.028 megabits which is roughly 3k per second under optimal conditions. This is really slow so video was not really something that you would have on a website. Even images were really slow where with large images you could actually see the image being loaded. This is why vector based animation systems actually made sense and why Future Splash was pretty impressive for its time. Instead of sending the bits that make up the image, vector systems would just send the points and edges that make up the shapes. For simple shapes, like those in early internet animations, this was significantly less information than the bitmap representation of the image so could be transmitted much faster.
Future Splash was competing with a technology that Macromedia was pushing called Shockwave which was acquired by Macromedia when they purchased MacroMind for their Director game making software. Macromedia acquired Future Wave, the company that was behind Future Splash, so they could add the vector animation tool to their shockwave platform. This is the reason why Flash binary files have the swf extension which stands for ShockWave Flash.
While I think Java could have taken control of the browser, this ultimately failed due to a conflict between Microsoft and then Java-owner Sun Microsystems. Sun wanted Java to be a standard that would work the same on all browsers while Microsoft wanted to be able to make Windows the operating system of the internet and added extensions to the internet explorer version of Java that made it work better on Windows due to the ability to access Microsoft APIs such as Direct X. A court battle ensued, Java on the browser stagnated, and alternatives for Java started being looked for.
This is roughly the time that ActionScript was added to Flash. While not the greatest scripting language, this made the product useful for games and other interactive web content and ultimately replaced Java as the go-to technology for interactive web sites. I started using Flash and wrote the original Flash Game Development book that this book is based on and everything was good with the exception of the occasional exploit, too many sites having intro movies created in Flash, and many extremely annoying interactive ads which happened to be written in Flash though would have also existed in any other alternative had there been an alternative with a large enough market share.
Flash was still a very strong standard when Adobe bought out Macromedia. Unfortunately for Adobe, things were about to change thanks to a fruity company releasing a phone a couple of years later. While saying that iPhone not supporting Flash was the cause of its demise is not the full story, it was certainly a big factor. The idea of plug-in for browsers was starting to be looked on as a negative factor as plug-ins were increasingly the target that hackers used to attack browsers.
The ultimate nail, in my opinion, was the release of HTML5 and the Canvas API (which if I recall correctly was an API designed by Apple for their browser). This allowed for web developers to do everything that Flash could do right within the browser knowing that all users with a browser could run the program. Granted, the HTML5 apps were slower and larger than the equivalent Flash files, but they worked without any plug-ins.