What you have is a valid ADO.NET connection string - but it's NOT a valid Entity Framework connection string.
The EF connection string would look something like this:
<connectionStrings>
<add name="NorthwindEntities" connectionString=
"metadata=.\Northwind.csdl|.\Northwind.ssdl|.\Northwind.msl;
provider=System.Data.SqlClient;
provider connection string="Data Source=SERVER\SQL2000;Initial Catalog=Northwind;Integrated Security=True;MultipleActiveResultSets=False""
providerName="System.Data.EntityClient" />
</connectionStrings>
You're missing all the metadata= and providerName= elements in your EF connection string...... you basically only have what's contained in the provider connection string part.
Using the EDMX designer should create a valid EF connection string for you, in your web.config or app.config.
Marc
UPDATE: OK, I understand what you're trying to do: you need a second "ADO.NET" connection string just for ASP.NET user / membership database. Your string is OK, but the providerName is wrong - it would have to be "System.Data.SqlClient" - this connection doesn't use ENtity Framework - don't specify the "EntityClient" for it then!
<add name="ASPNETMembership"
connectionString="Data Source=MONTGOMERY-DEV\SQLEXPRESS;Initial Catalog=ASPNETDB;Integrated Security=True;"
providerName="System.Data.SqlClient" />
If you specify providerName=System.Data.EntityClient ==> Entity Framework connection string (with the metadata= and everything).
If you need and specify providerName=System.Data.SqlClient ==> straight ADO.NET SQL Server connection string without all the EF additions