Difference between new and classic storage accounts in Azure
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). 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).
You can't convert a classic storage account (or any classic resource) to a newer type. You don't really need to anyway, unless you're trying to mix resources from classic and new, such as adding ARM-based virtual machines to a classic-based virtual network, or spin up an ARM-based VM from a vhd image sitting in a classic storage account (and for that example, you could always just copy the vhd to a new storage account). Note that, for general storage usage (blobs/tables/queues), you just need the URI and the primary (or secondary) key. With those, you can access your storage resources from anywhere, from any VM/website/etc, regardless if you're accessing storage from classic or new virtual machines, for example.
Check out this link for a general list of differences between classic and new resources.
One advantage of the new over the classic storage accounts is Storage Service Encryption (SSE):
Q: I have an existing classic storage account. Can I enable SSE on it?
A: No, SSE is only supported on Resource Manager storage accounts.
Q: How can I encrypt data in my classic storage account?
A: You can create a new Resource Manager storage account and copy your data using AzCopy from your existing classic storage account to your newly created Resource Manager storage account.
There is now a way to migrate Classic resources to the new ARM model. I've done a few myself and it worked as expected. Here's a guide from Microsoft: