Even Metal Gear fans will admit the series history is really hard to nail down, so let's make it clear right now: The Phantom Pain's Snake is Big Boss, aka Naked Snake in Metal Gear Solid 3, the star of Peace Walker, father to Solid Snake, and the villain of the original Metal Gear game. It's set in 1984, 11 years before Solid's first battle with Big Boss. In The Phantom Pain, Snake is just trying to rebuild.
According to early sources, his nickname for MGS5 appears to be Venom Snake, likely because he's filled with venomous rage in a quest for revenge on the people that killed and tortured those closest to him. He plans to do a lot of damage, but he's got to do a lot of rebuilding first. In fact, Snake himself has been rebuilt this time, including a newly acquired cybernetic arm and some very different gameplay.