Classic storage accounts are created using existing Service Management API's (the REST API stack that's been available for the past several years).
The newer storage accounts are created with the new Azure Resource Manager (ARM) API's (which are also wrapped in PowerShell and CLI now). One advantage of the new over the classic storage accounts is Storage Service Encryption (SSE) (to automatically encrypt your data when it is persisted to the cloud).
Ultimately they provide the same resources to your apps, but they're created/managed differently, and there are a few nuanced differences (such as the ability to tag resources that are created via ARM scripts).
