Sorry to pop in here, first post in fact but I may have the answer, is it not because Enfield FC folded and Enfield (18xx) is a competely new club (despite the name) like Enfield Town and FC United were, and so can't join in the FA competitions for a season. I don't know Barking's situation last year, they may still lay claim to being the same club, same history etc which may have made the difference.
Or I might be wrong, often am.
AFC Hornchurch and AFC Telford were formed before the April deadline for applying for entry into the next seasons FA Cup.
The New Enfield and AFC Wimbledon were formed after the entry date so miss a year.