Ryan's documents should explain the difference, but if you want the hip pocket class broken down shotgun style, then the biggest difference is what they do.
APM = Application Performance Monitoring, which is where the agents are reporting in on what your application is doing internally (calls it's making, length of those calls, application endpoints, load times, classes and methods used, how much memory it's using on your servers, etc.)
EUM = End User Monitoring, which is what your application does from an End User's perspective (tracking how long it took your web page to load, what happens when they click on a certain button, response times, heartbeat checks, and google analytics type information like browsers used, geo location, IP addresses, and OS)
