Questions of this type are usually asked of schoolchildren doing physics for the first time, and they ignore air resistance. If there was no air the bullet's downward force would match its upwards force.

You add the forces (bearing in mind they can be in opposite directions), and any force that is left over goes into accelerating the object.

If you drop something from being stationary, initially it starts to fall with acceleration g (9.82 m/s/s). But as it goes faster, the air resistance increases in the opposite direction to gravity. Eventually the forces are in equilibrium and it does not accelerate any more, so it has constant speed. That is called its terminal velocity.

When you throw something upwards, the air resistance and gravity are added, because they are both acting on the object downwards. So the upwards velocity is reduced much more quickly than due to gravity alone.

In the absence of air resistance, the up and down trips are nicely symmetrical, and if there is any horizontal velocity you get a perfect parabola. But with air present, the frictional energy loss makes the times and speeds for the up and down parts of the journey quite different.

Yes, the 'total force' to launch the ball is the same as catching it on the way down [ignoring air resistance]. But you can spread that force out over time. So you can have a very short but very great force [like catching the ball with stiff arms], or a smaller force over a longer time [like moving your hands with the ball is it comes towards you]. Your choice! Watch how people catch hard cricket balls: it is always the second way, because you get a lot less injuries that way. What I called the 'total force' is more correctly called the impulse. The speeds immediately after launch and immediately before catching will be the same [ignoring air resistance, and providing the two sites are at the same height].

Just to add:

A bullet out of rifled barrel spins. The point reduces air resistance. 

On the way down the bullet would tumble, increasing air resistance.