The best audio advantage with Linux is that you have full control over configuration of the kernel, drivers and applications minimizing all the crude that Windows and Mac OS has to support. That doesn't mean Linux will play audio better than Windows and Mac OS, just means you can improve boot time, free up memory and perhaps wasted CPU cycles and keep everything focused on the audio. Another bonus with open source software that I like is the ease of reporting issues and you can participate in development and test. With any computer audio system, the details are in the setup and even though you will have more control with configuring Linux, you have a greater chance with hitting a hardware licensing issue with drivers that might reduce functionality or features, so you need to do your research upfront.
Sadly though, most find setting up Linux extremely difficult and just selecting a desktop distribution like Fedora or Ubuntu (mint is Ubuntu based) you aren't much better off than Mac OSX or Windows. You really want a distro that gives you the flexibility to customize how packages are compiled and what is installed. One example, for a headless playback you don't need to install a window manager or any of default applications that go along with it. Avoiding the graphics drivers and window manager you can reduce the overall footprint of the OS saving you GBs of storage and even reduce the amount of RAM you need. It's not hard to make a 512MB raspberry PI with 4GB of storage and a USB DAC (or SPDIF) sound very good.
With that said, how big is the difference between Linux and Windows, there's no question that I can configure Linux to outperform Windows and Mac, in audio quality, overall usability and costs. My DIY recommendations though would be to select a mac mini and an OS you are comfortable with or willing to learn.
Jim