Most of the times they are interchangeable terms. Directory is a classical term used since the early times of file systems while folder is a sort of friendly name which may sound more familiar to non-technical users.
The main difference is that a folder is a logical concept that does not necessarily map to a physical directory. A directory is an file system object. A folder is a GUI object. Wikipedia explains the folder metaphor this way:
The name folder, presenting an analogy to the file folder used in offices, and used originally by Apple Lisa, is used in almost all modern operating systems' desktop environments. Folders are often depicted with icons which visually resemble physical file folders.
Strictly speaking, 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. It is comparable to a telephone directory that contains lists of names, numbers and addresses and does not contain the actual documents themselves.