As of today, there's no automatic way of converting a "Classic" storage account into "Azure Resource Manager (ARM)" storage account. Today, you would need to copy data from a classic storage account to a new storage account.
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Having said that, there's no difference in how the data is stored in both kinds of storage accounts. Both of them support connecting via account name/key and/or shared access signature. The difference is how these storage account themselves are managed. In ARM storage accounts, you can assign granular role-based access control (RBAC) to control what a user can do as far as managing the storage accounts (like updating, deleting, viewing/regenerating keys).
Regarding your question about using new storage accounts with ML workspace, I don't think it's possible today (I may be wrong though). Reason being, ML is still managed via old portal which doesn't have the capability to manage ARM storage accounts.