The Mk14 EBR was designed for the Navy SEALs and designed for high rates of fire. The M39 EMR was designed for the USMC and is designed to replace the DMR (Designated Marksman Rifle) offering more accurate and potent accuracy than the M14/M16/M4 rifles. Both rifles are built off of the M14 frame, but there are several differences.
The Mk14 EBR is manufactured out of a lightweight alloy, and it is spec'd with an 18" barrel, a weight of less than 12 lbs, and an overall length of 35". Thus it is a relatively light weapon capable of providing intense fire.
The M39 EMR is spec'd with a 22" barrel, a weight of over 16 lbs, and an overall length of 44". This rifle was issued to offer accuracy at relatively longer range than the standard combat issue setup. The 7.62mm round also has more punch than the 5.56mm round that the M16/M4 rifles are chambered in.
To conclude, the difference is mission requirements. There are more differences, but the main ones that you would immediately notice is that one is light and quick, one is heavier but more accurate than average.