I installed LocalDb using the SqlLocalDb.msi package and I can connect to it using SSMS using the server name (LocalDb)\\v11.0. So far so good. The problem is t
Use (localdb)\MSSQLLocalDB. That is the LocalDB instance intended for applications, independent of Visual Studio version.
Disregard my original answer: "With SQL Server 2014 Express LocalDB, use (localdb)\ProjectsV12. This works in both Visual Studio 2013 and SQL Server 2014 Management Studio." While ProjectsV12 will indeed give you a LocalDB instance, it's the wrong one, intended for use by SQL Server Data Tools.