I finally understand what happened.
When I upgraded to Azure SDK 2.0, it updated only the Azure web role project, not the web applications projects, which were still referencing the SDK 1.7 DLLs.
But no one noticed and it worked for a while. Lukos is probably right : Microsoft made some changes to their VMs, and the GAC now only contains the DLLs for 2.0 (since my deployment is targeting 2.0).
I made the necessary changes to my web applications (switching the DLLs) and now everything is fine.