As in the previous series of the game endurance effects on resistance to radiation (and possibly to poison), health, recovery rate, critical damage resistance and, most likely, some other characteristics.
As in Fallout: New Vegas (unlike Fallout 3) in Fallout 4 will be necessary to monitor the character (periodically to eat and drink). Whether it is implemented in a separate Hardcore mode or in the main game is not known yet.
Also the endurance will add in Fallout 4 a sprint mode. This is faster mode of transportation compared to running, reducing the reserve forces, and also allows you to shoot down the opponent to the ground, which can be achieved by adding the sprint mode using mods in the previous parts Fallout.