What is the difference between a directory and a folder?

Check "The folder metaphor" section at Wikipedia. It states:

There is a difference between a directory, which is a file system concept, and the graphical user interface metaphor that is used to represent it (a folder). For example, Microsoft Windows uses the concept of special folders to help present the contents of the computer to the user in a fairly consistent way that frees the user from having to deal with absolute directory paths, which can vary between versions of Windows, and between individual installations. ...

If one is referring to a container of documents, the term folder is more appropriate. The term directory refers to the way a structured list of document files and folders is stored on the computer. The distinction can be due to the way a directory is accessed; on Unix systems, /usr/bin/ is usually referred to as a directory when viewed in a command line console, but if accessed through a graphical file manager, users may sometimes call it a folder.

A folder is not necessarily a physical directory on a disk. It can be, for example, the printers folder or control panel folder in Windows.

Raymond Chen explains:

Windows 95 introduced Windows Explorer and along with it the term folder. What is the relationship between folders and directories?

Some people believe that Windows 95 renamed directories to folders, but it's actually more than that.

Windows Explorer lets you view folders, which are containers in the shell namespace. Directories are one type of folder, namely, folders which correspond to file system locations. There are other types of folders, such as Control Panel or Network Neighborhood or Printers. These other types of folders represent objects in the shell namespace which do not correspond to files. In common usage, the term virtual folder has been applied to refer to folders which are not directories. In other words, we have this Euler diagram:

(Virtual folders = Folders − Directories)

In general, code which manipulates the shell namespace should operate on folders and items, not directories and files, so as not to tie themselves to a particular storage medium. For example, code which limits itself to files won't be able to navigate into a Zip file, since the contents of a Zip file are exposed in the form of a virtual folder.