I'm currently using Lighttpd on a VPS. Before that it was nginx, but I came back to Lighttpd because I needed CGI.
Apache's HTTPD is mainstream. All documentations include examples for Apache and most modules are only available for Apache.
Is the effort to use something different from Apache really worth it?
That depends on your needs. I use nginx when it does what I need, which it usually does. Apache supports to many things that I don't often use. It's nice to use only what's needed.
I wouldn't consider it as premature optimization. Those two webservers are good products and strong in static content. Just because Apache is widely used, doesn't mean other products are just 'optimization'. In my experience lighttpd and nginx are easy enough to configure and you can use fcgi bindings for most programming languages.
You should consider the functionality you need and pick a server which offers those, possibly the one which uses less resources which most likely won't be Apache.
Yes, if you need something like 10k concurrent connections, you have a chance with LigtHTTPd but not with apache I think.
If it's worth switching while developing, I'm not sure. Probably use the webserver you know best for developing and optimize it later.
It really depends on your set of requirements. Yes, premature optimization is evil, but to be honest I don't really see how this can be it.
You would have a certain set of requirements from this webserver - maybe something to do with how easily it's installed? Portability? Featureset? etc. that dictate most of the decision for you.
I would say, don't worry about premature optimization with regards to which web server you choose - rather, objectively look at what you want it to be and what you want it to do, then make your decision from there and build your solution.
If you're running Rails, using Lighttpd is premature deoptimization. :) So it all depends on what you need. Your IT folks, if/once you have them, will thank you for using apache if you can.