The Project Assignments feature in CORE allows you to assign specific resources (employees or vendors), activity items, and expense items to a project. This restricts the drop-downs to the assigned items when creating time and expense entries. Project Assignments allows faster data entry by reducing the number of items in the drop-downs and also help prevent your from selecting incorrect activity or expense items. Check CORE Help Center for details.
On the other hand, the Allocation feature allows you to assign tasks (activities) and expenses to your employees for various projects with a stated duration of time. This enables employees to work with a budget and deadline on projects, and gives them an idea about what is expected of them. Check CORE Help Center for details.