What do you like best?
Nginx does what it does very well. It's very fast (I've used it just as a proxy server before), handles concurrency well, and is super configurable. At work, we spin up servers often and we want to be able to easily host multiple sites on a single server. By setting up localized nginx configurations, we can be up and running without having to hardly tweak the main server at all.
What do you dislike?
Occasionally there will be issues that are easily fixed by adding something to .htaccess. Since Nginx doesn't have that support, I have to find a workaround, but thus far that's the only thing I can find that I dislike.
Recommendations to others considering the product
If you have never used Nginx and you're hesitant to throw it into a big project, try using it on a side project first. Trust me, you won't be disappointed.
What business problems are you solving with the product? What benefits have you realized?
We use it primarily as a proxy server to direct all our sites through. It's been incredibly easy to get up and running, even across multiple servers, and is a true pleasure to work with.