The 7.62 Nato (also known commercially as the 308 Win) round was developed by the US military, and adopted by both the US military & NATO in 1954. It is basically a shorten 30-06 bottleneck type round. As it's Nato designation states, the cartridge case (brass) is 51mm long (actually 51.18mm), and it fires a 145-150-grain, 0.308" diameter bullet at approx. 2,750 fps. It was initially chambered in the M-14 service rifle and M-60 machine gun.
The 7.62x39mm was developed by the Soviet Union during WWII, to counter the German 7.92x33mm light assault rifle cartridge. It's a tapered bottleneck case, which requires the use of a curved magazine. Cases are usually lacquered steel, the length is 39mm long (actually 38.81mm), it fires a 123-grain FMJ 0.311" diameter bullet at approx 2350 fps. It was initially chambered in the Model 43 (SKS) rifle, then later in the AK-47. It is roughly equivalent in power to a 30-30.
The 7.62Nato is a far superior to the 7.62x39mm in accuracy, and power.

No, they are not the same.
The NATO is 7.62x51mm with the place of origin being the United States. 7.62x39mm was created by the Soviet Union.
7.62x51mm NATO was for the M14. The 7.62x39mm was for the AK-47.
The 51mm and 39mm measurements are in reference to the case length. See the below links for further information.
https://en.wikipedia.org/wiki/7.62%C3%9751mm_NATO
https://en.wikipedia.org/wiki/7.62%C3%9739mm