WebDAV:

• Adds 7 new HTTP methods
• Natively understands folders as "collections"
• Clients manipulate and control namespace
• Clients can "parse" URIs to apply semantics of hierarchy
• Supports locking of entire hierarchies
• Natively supports relocating and copying document to a different folder
• Does not separate modifiable resources from read-only view of resources
  

AtomPub:

• No new HTTP methods
• Natively understands listings as "feeds" and modifiable listings as "collections"
• Servers control namespace, clients can make suggestions.
• Clients can navigate to URIs identified in hypertext but not construct these URIs
• Natively supports only optimistic concurrency control of single resources
• Separates view-only resources from their underlying modifiable resources.
  

Bottom line is "use WebDAV if you can". AtomPub is a simpler technique, but it is also less feature rich. Of course, extensions and other such things will add back some of the features over time, if people like the lightweight foundation of AtomPub.