I have done some work with Apache, but as my workplace is mostly a MS shop, most of my work has been with IIS.
My observations.. IIS is actually easier to administer as long as your site doesn't get too complicated.
It can also depend on what you are more comfortable with writing the code for your webpage in. If you are looking at ASP, IIS may be a better choice. Apache, you are looking at using a component like Chilisoft ASP (or others) to use ASP code. Also, if you are looking at writing anything using .NET - you will have to use IIS as I don't think Apache can use .NET (yet). I will also say that IIS6 is a big improvment over previous versions but it will only run on Windows 2003 Server.
I have not worked much in the (fairly new) Apache 2.0 releases as most of my Apache work is done in the 1.x releases. But from what I've heard and read, there isn't too many complaints with Apache 2.0. Apache running on linux you have little to no startup software costs. All you would need to pay for would be hardware and bandwidth. IIS you would need to buy the Windows 2003 Server license (well, at least the win2k3 Web Server edition). And nothing really beats Apache and Unix variants for running Perl or PHP.
All that being said, if I had a choice here at work, I would be running Apache.
Apache 2 is as sweet as i don't know what. PHP is easier to write code in than ASP (Yes, i have written code in ASP) and Apache is a stable server that can handle a ton of load where i ahve seen IIS fail.
Personally, I prefer Apache. I have been fortunate to use both IIS and Apache 1.3/2.0. Apache is far more secure. IIS is easier to use as long as you website is pretty basic. But, then again, all of my servers (with exception to my PDC) are Linux/MacOS so I don't really need IIS anyway.
Don't blame IIS. It is a fully capable webserver when administered corectly. Apache is more powerfull (my experience). The reason IIS gets a bad rap is that it runs on windows (I know the irony of that statement). The trick to using IIS is to have a solid firewall in front of it. This firewall will proxy all requests and fillter out all malformed urls and known attacks. If you don't want to do this, then apache is a better choice. In the end, both can handle the same load, if configured properly. Also keep in mind, apache can run on windows, so you are not stuck to a certain OS when chosing it.