If you're just hosting basic pages without dynamic content any server will do - go with whichever you want.
If, however, your site has dynamic content or you want to add it down the road your choice in server will dictate your choice in development language.
I prefer Windows servers as everything I've found that you can do on Linux you can do on Windows, but you CAN'T go the other way around. Perl, PHP, ASP, and ASP.NET all work on Windows - on Linux you don't get ASP or ASP.NET (and don't say ChilliASP because we won't even go there...)
There are some who believe that one OS is better than the other, but the truth in the matter is that it really depends more on who ELSE is on your server - an over-crowded Linux server will perform just as poorly as an over-crowded Windows server.
As for security, again this is more about your host than the operating system. If you install EITHER operating system out of the box and leave it alone, you have more security holes than you can count on two hands - a properly configured Linux server is just as secure as a properly configured Windows server. Budget hosts tend to fail in this area more than the established (and more expensive) hosts.
Your host should also have a firewall - not some software crap mind you, but a real external hardware-based firewall like a Cisco PIX. Nothing but ports 80 and 21 (and 22 if Linux) should be open to the web server - if they don't have a firewall, SWITCH HOSTS because you're wide open. Unless you don't care about security :)
If you're comfortable that your host knows what they're doing in the security area then it really comes down to development languages and personally I find Windows servers to be superior in this area. But I also do 100% of my own authoring and don't run third party scripts - some scripts only work with a specific type of server (ie; some PHP scripts only work with Linux while others will work with both) so that's something to consider too.
I'm not trying to confuse you more, just give you all of the information ... many people here have an anti-windows agenda (as illustrated in the "don't let microsoft control everything" post) and that doesn't help you make an informed decision about what's best for YOU.