What do you like best?
The easy installation, easy of searching out of the box.
What do you dislike?
querying is not easy. If you want to understand the underlaying querying, analyzers and tokenizer, it should take you some time. This is not trivial.
Recommendations to others considering the product
If you want to enable searching capabilities to your apps, elastic is easy to setup, and if you want the standard out of the box capabilities - you should be ok. If you need more advance querying from elastic, make your self some time off to learn.
If you need logging monitor for your server, this is the right tool as well, combining it with logstash and kibana
What business problems are you solving with the product? What benefits have you realized?
Searching in a huge amount of data, was made easy with elastic search. It provides a search capabilities to our applications.
In addition, it solves us the ELK stack (Elastic, Logstash, Kibana) - which provides us monitoring for logs in real time.