Na maioria das vezes, são termos intercambiáveis. Diretório é um termo clássico usado desde os primeiros tempos dos sistemas de arquivos, enquanto a pasta é um tipo de nome amigável que pode soar mais familiar aos usuários do Windows.
A principal diferença é que uma pasta é um conceito lógico que não necessariamente mapeia para um diretório físico. Um diretório é um objeto do sistema de arquivos. Uma pasta é um objeto GUI. Wikipedia explica desta forma:
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.