A body projected horizontally moves with the same horizontal velocity although it is under the action of gravity. why? ​

Hello friend!
There is a difference. The motion of a particle as projectile has two components - horizontal & vertical. The height of projectile above the ground is solely responsible by its vertical component while the horizontal displacement(range along x axis) is directed by its horizontal component.so, it is under the influence of gravity only when it has a vertical component in it. so there is no component of acceleration along the horizontal. That's why we take horizontal velocity constant in the motion at all points.
Regards.