Your question is specific to web developers, and I cannot comment on that.
In general, WebDAV is getting more popular from my observations. Thanks to services like MobileMe iDisk, many iPad/iPhone applications can speak WebDAV. WebDAV is also natively supported by Windows (XP/Vista/7), Mac OS X and also by Linux (quality of support in these native clients is another matter). This gives you a nice collection of clients.
On the other hand, only real benefit of WebDAV protocol is that it is based on HTTP. It's not very efficient and can be indeed difficult to troubleshoot.