What concerns do the FAA and EASA have in 2014 with "own-ship position display" on EFBs?
Display of ownship's position has a very strong effect on the mental model that the pilot has of it's navigation. When the EFB displayed position is wrong, the change that the pilot will detect that by comparing to other instruments or outside visual are very limited. This is a safety concern. 

To have the EFB approved to display ownship's position, the most straightforward way would have been to certify it under TSO C129a "Airborne Supplemental Navigation Equipment using the Global Positioning System (GPS)" TSO-C129a has now been superseded by TSO-C196 which has slightly different requirements.

Often class I or class II EFB will have a GPS receiver that technically could be used as the source of ownship's position. The GPS receivers chips used in are not designed and certified against the "Minimum Operational Performance Standards for Airborne Supplemental Navigation Equipment Using Global Positioning System (GPS)” (RTCA document DO-208), which is a requirement of TSO-C129a or "Minimum Operational Performance Standards for Global Positioning System/Aircraft Base Augmentation System" (RTCA document DO-316) which is a requirement for TSO-C196.

The major technical difference between simple commercially available GPS chips and aviation grade GPS chips is that the latter include a Receiver Autonomous Integrity Monitoring (RAIM) function which does Fault Detection (and Exclusion).

If a GPS satellite would drift out of it's orbit or GPS data gets corrupted in the receiver, an ordinary GPS chip would most likely display a position error, while an aviation GPS chip would either indicate a failure state (fault detection) or still display the correct position (fault exclusion). This protects the pilot from being mislead by a map shift.