Filter expressions allow more flexibility by allowing you to determine who can see an entry based on an expression - so for example you could say 'if field x is on the document, and its value is y, only members of group z can see the document'. Note that filter expressions only control seeing the document, not anything else.

The reason it's generally recommended as a 'last resort' is that all these calculations have to be performed on the fly, every time any user would be accessing the document or the containing folders. This isn't usually too big a deal on browse (although it can be), but it can really add up on search. Think about having to decipher and check calculations like this on every single entry throughout a large search result before returning the results to the user.