Hi Rob,
On your title, as you are mention "Automic - Enterprise Scheduler -v- Applications Manager" I am guessing you might be referring to Automic flagship product - Automic Workload Automation (AWA). This is what other companies who were on Applications Manager ended upgrading to.
There are a few that already come to my mind, such as the sheer scalability of it compare to AM - in which you are not bottle neck to the process such as the AM "Master" and the number of jobs you run/queue up/have out on the backlog... An Active-Active setup.
There also easier import/export method, version management/roll back... along with that in higher version of AWA there are the Zero Downtime Upgrade (for the system and agents)...
Overall AM vs AWA, in term of functionality AWA allow you to expand what you are doing now + more/or expanded functionality. The questions is.. Is there anything particular now with AM that you are looking for the new Scheduler to have?
In additions, I will move this post to the Space/Section. To better reach the audience who are using our flagship product (AWA) and were on Applications Manager. As they may have more input (since I just mention a bare/brief on a tiny amount of things) to your general inquiry.