What are the advantages of using UNIX for a large network compared to something like Windows Server?

Unix began as a multi user, networked, system, and has built upon that knowledge since the dawn of the UNIX Epoch.

Windows began as a single user, non-networked, system, and has only recently begun to embrace some of the security protocols which *nix systems employ, although Windows often poorly implemented those protocols, seemingly as an afterthought, as illustrated by the multitudes of Windows vulnerabilities and exploits.

The majority of the internet runs on *nix based systems because of the benefits outlined above, and more, such as the previously mentioned cost factor.

Then there's the open source paradigm, which allows developers to respond more rapidly to bugs and security concerns, as opposed to a closed source system controlled by corporate interests whose main concern is profit margin.