Answer:
The similarity is that they both follow the laws of physics.
In both cases, the motion is opposed by the medium where the object presses through the medium and forces it to part and flow around the object. This creates friction.
If an object is smooth and pointed like a bomb, the friction is minimised. If it is rough with pockets or vertical flat surfaces, the friction is maximised. A parachute presents a round hollow cup face to the flow, maximising friction. This is due to a hole in the top of the parachute.
The difference is the power source. In the vertical direction it is generated by gravity, the force created by the planet. This force is opposed not only by friction but also the difference between the density of the object and the medium surrounding it. If there is a large difference then the acceleration will be large and a high percentage of g.
For example we have a density of 1000 kg/meters cubed and air has a density of 1 Kg per meters cubed, the difference is 999, so we fall at an acceleration of 999/1000 x g. minus the effects of friction.
In the horizontal direction, gravity still opposes the movement creating friction in the vertical direction such as dragging a heavy object on the floor behind you. Force needs to be created horizontally to oppose the friction and to create an acceleration. A balance is soon reached where the acceleration stops and the object is dragged at a constant velocity.