The advantages of one and the other monitoring services are server level monitoring (CPU, Mem, disk usage), application level (sends information to your web application and awaits a specific response), or both.
As mentioned earlier, you must have both. If your application works well, there is a chance that a warning about the system level will be caused by high traffic traffic, etc., and you can fix the problem before your applications suffer. But application level warnings are the most important, as they usually check what the user sees, and can warn you when any part of your web stack has a problem (apache goes down, dead links, broken inserts, database corruption, etc. etc. etc.).
Complex services such as New Relic (NR) can test both.
Regarding your second question (is it worth it), my opinion is that it depends. I have been using the new relic for several months and find it very useful. Especially when determining why a problem arises, and not just getting a warning. Warnings are complex, and NR can monitor both the application and the server itself. In general, I believe that NR is an excellent product for full monitoring of servers and applications, but it is expensive. But for a small server (for example, mine, which receives only 2000 visits per day on 10 small websites and gives a small income), I do not find an excuse for the cost. I plan to return to the free / more affordable service for basic alerts and leave troubleshooting problems.
For free basic services, I use CloudKick a free developer plan for server monitoring and a less complicated / expensive pingdom.com for 1 free application monitor.Pingdom alerts are not so complicated, but it monitors and alerts at the application level at reasonable prices.
My suggestion would be if your uptime is critical, then you should include a sophisticated monitoring system like NR in your budget. If your uptime is not critical, then a few free services can do the trick. But in any case, you should keep track of your server and applications.