Yes you can restore a backup file which is taken from SQL Server Enterprise Edition in Standard Edition but if the source database is not using Enterprise Edition specific features, like partition functions in our case, it could be a problem.
This happened to one of my juniors when he was restoring a database (on SQL Server 2008 R2 Standard Edition) from full backup, which was taken from SQL Server 2008 R2 Enterprise edition. A 6 GB database backup file took more than half an hour to restore and after showing 100 percent completion, on starting database, it returned FAILURE ERROR.
Database 'SQL2008R2_Ent_PartitionFunction' cannot be started in this edition of SQL Server
because it contains a partition function 'SSF_PF_Right'.
Only Enterprise edition of SQL Server supports partitioning.
Database 'SQL2008R2_Ent_PartitionFunction' cannot be started because some of the database functionality
is not available in the current edition of SQL Server. (Microsoft SQL Server, Error: 905)
Basically the restore process is completed successfully and you can see the database name in the list and its files on required location but in inaccessible mode.
I think Microsoft should improve this process and these prerequisites should be checked first.