Answering my own question after more investigation:

It seems that WebDAV is more like file sharing then just a file transfer protocol like FTP. I.e. it is more like an AFP/SMB/NFS network file system where you can 'mount' a remote file system and the files/folders appear to all of your applications like a local file system. I guess in some senses it is like an open standards version of those proprietary protocols, plus more.

A file sharing model allows for direct editing of the remote content instead of an upload/download model. (Well, at least some portion of the data has to move up and down the wire in order to edit it, but it is transparent to the user and doesn't require a series of steps and helper applications). This enables a locking model so that when one person is editing a file, others are locked out from modifying it, again much like what would happen in a local or networked file system. That would be very important in a collaborative development effort - e.g. if more than one person was simultaneously maintaining the content of a web site.

If I read the docs correctly there are other advantages like you can tell the WebDAV server to copy a file/folder from one location to another on the server - as opposed to copying the data up to your local machine and then copying it back down (which I believe would happen if you use AFP/SMB/NFS).

There also seems to be some high level support for things like server side searching and querying of meta-data about the files on the WebDAV server, like getting the mime-type of a remote file etc.

If my interpretation of the readings are correct I can see how those things would be a nice set of advantages over FTP, and with the extra server side features, an advantage even over a simple network file system. Most of what I do is from Mac to Mac so I generally use AFP if I want to mount remote file systems. But WebDAV looks like it will be more efficient network wise for many things since there is more that is done on the server side. There are likely other advantages, but I don't fully understand WebDAV enough at this point to extrapolate any more.

Overall, it looks like WebDAV is 'a good thing'. It looks like I'll spend the effort to get it installed and give it a whirl.

First it is much more firewall-tolerant - you will be able to transfer files even if you are behind a firewall, because DAV uses HTTP port by default and defaults itself to only one port instead of 2 (as FTP does).
Second - DAV connections are much more integrated into customer-type OS (mind there is no FTP folders by default in Windows and FTP mounting in OS X works in read-only mode, so you can downlaod but cannot upload, delete...). Third, DAV file transfers (especially small) are to my experience faster than FTP.
Third, DAV supports locks/check-ins and check-outs (which are useful in multiuser workflow, for project authoring where many files are used). Many today's content management systems are built around DAV.