Microsoft.WindowsAzure.Storage is version 2.0 of storage client library while Microsoft.WindowsAzure.StorageClient is the older version. There have been many changes in version 2.0 of the library (some of them are breaking). If you're starting new, I would actually recommend using 2.0 of the library as I found it more intuitive and easy to use than the older version. If you have an application which makes use of 1.7 version of the library, before you decide to upgrade, I would actually recommend reading the following blog posts by Windows Azure Storage Team:
However please note that there're still some components that your application might be using which has a dependency on storage client library 1.7. Windows Azure Diagnostics is one of them. So for some time you will need to use both versions. Good thing is that you can use both versions simultaneously in your project.
Hope this helps.
EDIT:
I also wrote a few blog posts about migrating code from storage client library 1.7 to 2.0 where I covered some basic scenarios. You can read those posts here:
Migrating blob storage code: http://gauravmantri.com/2012/11/28/storage-client-library-2-0-migrating-blob-storage-code/
Migrating queue code: http://gauravmantri.com/2012/11/24/storage-client-library-2-0-migrating-queue-storage-code/
Migrating table storage code: http://gauravmantri.com/2012/11/17/storage-client-library-2-0-migrating-table-storage-code/