A mail system can have folders "inside", accessible e.g. via web mail or native apps. This has nothing to do with how that mail system lets third party apps access messages.
If said mail system supports only POP3 for third party apps, there will be no distinct folders in those external apps.
If .... IMAP .... there will be folders.
POP3 and IMAP are not about how a mail system organizes messages internally, they's specific methods letting third party ("external") apps connect.
One example:
Hotmail / Outlook / Live has folders if you use their web mail, or the native app, but for third party apps, it supports POP3 (no folders) and ActiveSync (folders, but requires licensing from MS). IMAP is not supported (promised last August, not delivered yet, AFAIK).