This is usually a permissions issue since the two service ran under different users and groups
Usually gammu is ran under the gammu user and playsms is ran under the apache user (www-data on Ubuntu) you can run your gammu service under your apache service so the apache service will be able to write and read from the gammu folders as well as gammu. You can configure the user that gammu run on /etc/default/gammu.
Also you can make your apache user part of the gammu group. Try to limit the permission to the gammu folders to avoid security issues or set your gammu folders in a public area where there is no access to the system (be aware that other people will be able to read your sent folder).