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.