There's frequently a lot of confusion in the APM market -- As you might have noticed, there's a lot of overlap in what both tools can do. Graphite is primarily used for time-series metrics, whereas AppDynamics is a full-stack APM vendor.
In other words, Graphite can be used to aggregate any metric you're interested in (such as CPU usage), which you would then correlate with other relevant metrics (such as number of web requests/second). If you're specifically interested in tracking application-level performance (what is my website throughput or where are my endpoint bottlenecks), you would need to find an integration for your specific environment, or custom instrument your application.
AppDynamics also has graphing features, but aims to bundle any other relevant use cases an application developer or operations person might need. For instance, it includes service topology maps (see which services are running on your infrastructure, and are talking to which other machines) and application logs (to parse unstructured text). They have also built many integrations (to collect relevant metrics), usually for large enterprises.