Good answer by @MDMarra. Providing one other perspective, to round out the answer: You could roll your own backup, using Blob storage as your backup storage medium. You're right that there's a REST API, and there are also SDK wrappers for .net, Java, php, python, ruby, and node.js, as well as both PowerShell and node-based cross-platform CLI, all capable of manipulating storage.
Regarding Robocopy: Because blob storage has a REST-based protocol underneath, you'd need a tool slightly different. The Azure team published AzCopy to provide functionality very similar to Robocopy.
So... in theory you can create your own backup/restore scheme. This will have a cost footprint equivalent to the cost of Storage only. However: As @MDMarra pointed out, the Azure Backup Service provides a backup service, which is richer feature-wise than just file-copies: Compression, data-retention policies, incremental backups, etc.
One last thing, regarding throughput: An Azure storage account has a scale target of 10Mbps ingress/15Mbps egress, with total capacity of 200TB per account (and you can have multiple accounts). Given your need to move 1TB weekly, that should be no issue.