As in many other cases, they are almost synonyms. עסוק means I'm busy right now, I'm doing something. עמוס means I have a lot of task to accomplish. The עמוס has many tasks but not necessary right now, while the עסוק has one right now. In the case of the עסוק you should wait patiently till he finishes his task, and in the case of the עמוס come tomorrow or ask somebody else for help.. :-)

עמוס - مضخوط עסוק - مشغول