In his excellent and popular comparision chart, ErikEJ draws a distinction between SQL Server Express 2012, and SQL Server 2012 LocalDB. Ho
I use a table from ErikEJ, which shows: the features and differences between SQL CE 3.5, SQL CE 4, Local DB and SQL Server 2012. According to this table the differences between Local DB and SQL Server 2012 are:
Installation size:
SQL Server 2012: 120 MB download size; 300 MB expanded on disk
Local DB: 32 MB download size; 160 MB on diskRuns as Service
SQL Server 2012: Yes
Local DB: No (runs as process started by app)FILESTREAM support
SQL Server 2012: Yes
Local DB: NoSubscriber for merge replication
SQL Server 2012: Yes
Local DB: NoNumber of concurrent connections
SQL Server 2012: Unlimited
Local DB: Unlimited (but only local)
NB: Sorry this isn't "official documentation", but hope it's useful to the next bod as it answers the underlying query about the differences.