I think what Flash (the program) really excelled at was giving creators the ability to draw, animate and program interactive objects, all wrapped in a single neat package . I would contest that no other software, even today, has come close to being able to do these three so seamlessly at once (Really, I encourage you to try to name one!) Sure, when it came down to it you can say it had poor performance, security, fil…
It didn't have ActionScript yet, but if I remember correctly, it did have the ability to link visual elements to arbitrary frames within the animation timeline (as well as to other URLs), so it was actually pretty powerful (a bit like HyperCard for the web).
I was still in elementary school at the time, but I remember creating original animations that my friends all thought were produced by professional animators (or templates), and it certainly expanded my vision of what could be accomplished with a computer. The fact that you could deliver such rich (and interactive) graphics/animations in such a small package - over a 28.8 kbps modem - was like magic. Keep in mind, this was before streaming video became practical for anyone, and simple things like JavaScript[1] rollover effects on buttons still had questionable browser compatibility. Apart from tiny GIFs[2] (usually flames or "under construction" signs), some random tags and scores of Java applets trying to make text scroll across the screen, the web was largely devoid of motion. I would argue that FutureSplash was the first technology to show us that the web was capable of delivering high-quality motion graphics (things that looked like video) - i.e., content delivery that would eventually compete with television. Indeed, its successor (Flash) would go on to build off that idea by incorporating streaming video support (responsible for most of the early video content online).
Not only was the technology impressive under the surface, but as dyarosla mentions, the animator software struck a unique balance between being powerful and extremely approachable for beginners. That combination continued, for the most part, through Macromedia and Adobe.
[1] Strangely enough, I seem to recall a few years later (probably when it was under Macromedia's control) a player was released that could play Flash animations in pure JavaScript (eliminating the need for a plugin). It's interesting to think about how far we've come since then, yet how similar that was to our modern web tech stack, at least in concept.
[2] Actually, I don't think they were even animated GIFs as we know them. Client pull and server push animations: http://www.oreilly.com/openbook/cgi/ch06_06.html