SQL Server Express vs express localdb
LocalDB is a special, low impact version of the SQL Server engine, that is not installed as a Windows Service, but launched (made to run) on demand by the ADO.NET client opening a connection to it. It is intended for single user scenarios, and not for any production use - for production you should use SQL Server Express (or higher)
I have a blog post that compares the various engines here: http://erikej.blogspot.dk/2011/01/comparison-of-sql-server-compact-4-and.html