Yet another frustrating and confusing process for such a simple task.

Export contacts to a vcf file on the device:
1. On the S4, navigate to contacts (Phone -> Contacts).
2. Press the device menu button (bottom left device button).
3. Choose Settings -> Contacts -> Import/export contacts -> Export to device storage.
4. Confirm the transfer.

Pair the S4 to the S5 using bluetooth
1. Activate Bluetooth. Pair the devices.

Send the contacts vcf file via Bluetooth
1. On the S4, Choose Applications -> My Files.
2. Press the device menu button (bottom left device button).
3. Choose “Search” and type “Contacts”. If the contacts file is not shown return to the first step “Export contacts to a vcf file on the device”.
4. Long press on the file.
5. Choose “Share via”.
6. Choose Bluetooth -> Select the S5 device paired earlier.
7. Accept the file transfer on the S5.

The contacts should now appear on the S5.