The key to understand the problem here is the provider and in your case it is "SSL Provider" and the main problem is that connection time out. If problem was related with "TCP Provider" there could be several reasons related with network etc however in your case it is a security related issue within the connection framework.
Next step to investigate this problem is that on the same machine, you can connect from SSMS to same SQL Database however not from IIS which means the IIS application process is not able to get proper certificate to create the SSL tunnel to create the successful connection. It is possible that you have some log written System Event log so please check there.
Because you are using Entity Framework and depend on EF version there are some changed in the connection string so verify if that is what you have:
<add name="MyDatabaseModelEntities" connectionString="metadata=res://*/MyDBModel.csdl|res://*/MyDatabaseModel.ssdl|res://*/MyDatabaseModel.msl;provider=System.Data.SqlClient;provider connection string="data source=abcdefg123.database.windows.net;initial catalog=MyDatabase;persist security info=True;user id=MyDatabaseUser@abcdefg123;password=p@$$w0rd;multipleactiveresultsets=True;Trusted_Connection=False;Encrypt=True;App=EntityFramework"" providerName="System.Data.EntityClient" />
Finally if you are using EF 4.3.1 and connecting to SQL Database please have a look at this article and check if you have similar configuration.