As someone blessed with both Moulton APB and Birdy Monocoque, the Moulton gets my vote. Nice as the birdy is, the suspension and stability for the Moulton is quite far ahead. I recently nicked the light wheelset off my Swift and put them on the Moulton - wow what a major difference!
The Moulton is a nicer ride. I am currently commuting with it. And being steel, somehow (perhaps entirely unjustifiable) I feel it is a bike that will live longer.
The Birdy wins with folding of course, and its quality is impeccable. But I don't find the suspension all that good. I need to do some experiments involving the famous rubber doorstop method for improving suspension behaviour, inspired by Julian Edgar.