How does Azure Direct apply to an educational institution?
You mean in terms of prices and discounts? Direct Restore to Microsoft Azure is a free product, you should pay only to Microsoft for using Azure compute resources.
I'd also like to how replicating backups to Azure is different than using a cloud provide
Direct Restore to Microsoft Azure is not about sending data for long-term archival purpose to the SP of your choice, it's more about taking backups and restoring VMs present them to Azure IaaS (Virtual Machines) - approach suitable for backup verification, creation of test environments or migration.