WebDAV and Wiki get compared frequently. They're often in frameworks that solve the same problem, but they play different roles within those frameworks.

A Wiki is a Web site running Web application code that allows people to browse Web pages, easily add new Web pages, and easily edit any Web page through Web forms (the Web application code powers those Web forms and handles their input). Originally Wiki described the first such application but the word has spread and similar applications that run completely different code are also called Wikis.

WebDAV is an IETF Proposed Standard, RFC2518, describing a set of extensions to the HTTP protocol. These extensions allow a client application (or other software agent) to create new HTTP resources more easily, manage them, find them without following hyperlinks, and change them in a way that multiple authors can work together simultaneously.

Wiki uses HTTP forms to transport requests	WebDAV extends HTTP by defining new methods.
Wiki is a UI.	WebDAV is completely orthogonal to a UI.
There exist a number of different Wikis, implemented in different languages, supporting different features.	There are a lot of different WebDAV servers as well as WebDAV clients. The clients can theoretically all work against any server although in practice there are a few outliers (special purpose clients or servers) that don't interoperate so well with the main group.
Wiki is a thin-client application, where any browser can be the client because the smarts are on the server (the code running the Web forms).	WebDAV was created to allow rich clients to author Web resources. A WebDAV client is definitely the entity in control -- but you have to download or find WebDAV client software. Luckily, these clients ship now in all Windows and Mac OS's.
Wiki is a generic name that seems to be used if something is similar to original Wiki	WebDAV is a set of requirements, so a WebDAV implementation can be tested for compliance.
You'd probably not call something a Wiki, if it didn't enable multiple users and allow easy creation of new pages, and result in a Web site collection of human readable pages.	WebDAV can be used to do synchronize with a source code repository, or to change configuration files, or to access a calendar server, or many more things that you would never call "shared authoring".
It's easy to extend (a) Wiki, just code in the new feature -- e.g. search.	It's hard to extend the WebDAV standard, it requires IETF drafts, reviews, and many revisions in the IETF doc format. It can take years -- e.g. SEARCH, begun around 1998. OTOH it's easy to extend a WebDAV server -- e.g. Xythos added anonymous permission tickets to the WFS server -- but no other clients are likely to interoperate with a custom feature like this for a while.

Anyway, in this case, the different roles mean that Wiki and WebDAV are not at all incompatible, and to compare their advantages and disadvantages as if they are both apples can be misleading. Brad, whom I met at a party last week, realizes this: he added WebDAV support to an existing Wiki codebase. It allows another way to add pages to a Wiki or to modify them, and can interact completely smoothly with the existing WebUI. You'd think somebody would have done this before now, but I can't see that anybody has. Way to go Brad.

[UPDATE Aug 9: clarified "extend webdav" to differentiate between extending the standard, and extending a server]