For example, if I'm doing audio processing, wouldn't I just be better off using x86 since that has a larger install base?
"Larger" is a good word for the x86. It needs a larger space claim on a PC board, it has a larger price tag, it needs a larger power supplies (and larger batteries if that's how you're powering your app), etc.
DSP chips are specifically targeted at doing the number crunching that you need for signal processing. They tend to give up ease of programming and flexibility, but in return they do what they do comparably cheaply, fast, and at low power levels.
If you're contemplating an application where you're deciding whether to do DSP in the close vicinity of a PC -- yes, by all means, use the x86 that you'll find therein (or a GPU -- they're better for DSP in a lot of cases).
If you're contemplating an application where you're building something in a stand-alone box to do DSP then you may find a DSP chip to be more economical.
Note that there's a lot of "might", "may", etc., in my answer. That's because for any given application a DSP chip may or may not be "best". It's also because while DSP chips were pretty much the hands-down best choice for audio-speed applications 30 years ago, today the best embedded-space step down from a PC may involve a processor with an ARM core -- the ARM core's are not, by any means, purpose-built DSP chips. But they are getting more and more number crunching with every iteration, many are cheaper than an x86, and they're really good general-purpose processors in the bargain.