What do you like best?
Open source, high performance webserver (and beyond with the modules) that can scale large amounts of connections without an issue. A lot of the additional modules provide support for other features or protocols - and in the newest versions, modules can be dynamically loaded too now.
Nginx provides very fast support when used as the front-end webserver and proxying back to another one [locally], such as Apache for the specific applications that may require .htaccess to still work.
What do you dislike?
I'm not entirely sure there's a whole lot to dislike about it. Nginx is pretty much a drop in replacement for any other webserver using different SAPI's via fastcgi instead or modules. It does lack .htaccess support that is still ever so needed at times based on customer applications.
Recommendations to others considering the product
Seriously, you can't go wrong using nginx! As long as you have everything properly tuned accordingly, it should only increase performance and response time.
If you have the requirement of using .htaccess still, you can still run the site via nginx and just proxy it back to Apache.
What business problems are you solving with the product? What benefits have you realized?
We're able to scale wordpress extremely well using nginx+php-fpm+fastcgi caching for commonly requested pages, so the webserver doesn't have to re-process them if found in cache.
Nginx is also the webserver being used for BelugaCDN -- although the CDN proxy is written in Go.