You can also use AzCopy that is part of the Azure SDK.
Just click the download button for Windows Azure SDK and choose WindowsAzureStorageTools.msi
from the list to download AzCopy.
After installing, you'll find AzCopy.exe
here: %PROGRAMFILES(X86)%\Microsoft SDKs\Windows Azure\AzCopy
You can get more information on using AzCopy in this blog post: AzCopy – Using Cross Account Copy Blob
As well, you could remote desktop into an instance and use this utility for the transfer.
Update:
You can also copy blob data between storage accounts using Microsoft Azure Storage Explorer as well. Reference link