Some of these are general advantages, some are general advantages of desktop Linux, some are specific to webmasters.
1) You probably use Linux or Unix servers. Its useful to learn Linux better. Things like commands that are useful on your server, are often useful on your desktop too, and vice versa. Use *nix on both and you can learn more, and can apply what you learn more.
You can also run development servers without bothering with VMs etc.
2) Easy software installation. This is a big one. If you are anything like me you install software regularly, and not standard desktop apps.
There is a huge difference between Linux software installers and Mac style app stores. The Linux variety install dependencies. For example, if you install something written in Perl through the installer, it will check whether you have Perl installed or not, and download an install it for you if its not. If different versions are available it will install the right version!
3) Easy upgrades to new versions. Most desktop distros (the big exception being Mint), allow you to upgrade painlessly within the GUI to a new version. I believe you can now do this with MacOS (and Windows?), but it looks like the Linux version is still simpler (automatic upgrades to virtually all your apps, virtually nothing to check before upgrade,...).
4) Security. Whatever the reason (popularity, better design - happy to argue about it in another thread) Linux desktops are less likely to be compromised. It is very easy to have lots of critical information on your desktop (database and server passwords, for example). Security is a lot more important for a webmaster than someone who uses a PC to go online to use Facebook. Yes, I know Windows can be made pretty secure these day, but if you have to work hard to do it, its easy to slip up (and a waste of time).
5) Lots of development tools. Linux has an excellent ecosystem of development tools. Many of them are cross-platform, but many are not, and good many of those that are cross-platform work best on *nix.
One caveat: Linux does not have as wide a choice of WYSIWIG web design or graphics tools. IN particular no Dreamweaver or Photoshop for people who have spent years learning those particular tools.
6) Lots of support for managing servers and other common web master tasks. For example, most linux desktops will allow you to open the file manager, type in sftp://example.com/home/me and then act just as if it was a local directory - not just in the file manager but in most apps. You can use the file dialogue in your text editor and to open a file on your server.
7) Highly configurable. A choice of desktop environments, most of which themselves are configurable, means you can configure you desktop to suit your workflow and you preferences. Features like multiple desktops (which almost all Linux desktop environments support) support complex work flows.
8) No vendor lock-in. If Ubuntu starts being obnoxious, I can painlessly switch to another distro and use all the same applications.
This is also an advantage of the open source applications you use on Linux. Applications can be forked if developers let users down, and open formats make migrating your data to an alternative easier.
9) The *nix approach of combining lots of simple tools shines on the command lines, but not just there. Lots of applications work with/talk to/ each other.
Someone else think up number 10 please!