Scalability Unless you've got a very tightly scoped project, the inherent creep in data model will invariably lead to your data set growing. If you want to follow any type of normalisation of your data it means having multiple text files. This type of design is not going to scale very well.

Maintaince Adding records, removing records and maintaining the integrity of your record structure can be difficult. You've got two clients requesting access to the same record, both writing, which should go 1st? we're talking about locking. How do you manage backups? restore a complete file? what if its been updated? we're talking transaction logs..

Integrity What character do you use as a delimiter? and what if a user wishes to use it for a legit purpose? what about a data model which says only values 'a, b and c' are valid, and a user enters d? This kind of things can be handled very easily using foreign key constraints in an RDBMS.

Any serious designer will tell you an RDBMS is the best way to go in terms of building data models (assuming you're data model is relational). You've got all the access control, the transactions, the ability to fundamentally normalise your data, its all there, and, its all available freely.

Now all the negative stuff is out of the way, there are instances where a FF may be handy. Read only as MarkM pointed out is probably the best use of flatfiles.