What do you like best?
We love it's error tracing capabilities -- it lets us know when anything is broken without having to monitor logs, which our QA team uses extensively during testing / when looking into and reproducing production issues. When you're a paid subscriber, it provides extensive trace detail on exactly where your application is slow in production environments, without the performance tracing slowing down the application. It also gives you a good sense of what your customer is experiencing with your application without having to rely on customers complaining that things are slow or broken.
What do you dislike?
New Relic has moved a lot of features that used to be free into paid tiers. Their pricing structure is unnecessarily expensive because it's per server or per container pricing. Unfortunately, a modern application is bound to have several servers that host the same application behind a load balancer. Since we don't use containers, that forces us to decide whether we want the pro features, or want to limit our scalability; we wish the pricing was per-application instead. So far we're stuck on the free edition as a result.
Recommendations to others considering the product
Balance the size of your application and user base against their hefty per-server pricing. Their free offering is on its way out, leaving only the paid option. Consider whether you have the money to afford a pro subscription for the number of servers you currently have, as well as the for servers you expect to have in about 1-2 years. If you have the money, there's nothing out there quite like New Relic and it's worth the investment.
On the other hand, if you don't have the money on hand for that, going pro is going to limit your scalability, and the free edition is not going to provide you with the data you need. So, in that scenario, it might not be worth integrating into your application and you'll probably want to get by with something less capable and less expensive.
What business problems are you solving with the product? What benefits have you realized?
We needed a way to identify performance bottlenecks that might only show up in production environments without having to replicate the entire setup and load characteristics of production in a test environment. New Relic's ability to profile applications in production helps with that. We also needed a way to easily triage issues that happen in production environments, which New Relic's error tracing and error analytics does natively. That significantly reduces the amount of research we have to do when something is not working / customers report issues.