Q:

Why don't we use rapidity instead of velocity?

In school we learn that we can add velocities together, and then later on we learn that it's not correct and that there is a speed limit. Why create all this confusion when we could just use rapidity to begin with?
Rapidity is defined as $w = \mathrm{arctanh}(v / c)$, where $v$ is velocity and $c$ is the speed of light in a vacuum. Rapidities can be summed and have no upper bound. At non-relativistic speeds it acts proportional to velocity.
In fact, at non-relativistic speeds, we could substitute $v$ for $wc$ (rapidity times speed of light), and one could hardly tell the difference. The ISS moves rather fast at a velocity of 7660 m/s (27,576 km/h), and has a $wc$ of about 7660.0000016667 m/s. Why can't we just substitute velocity for rapidity in real-world and classroom use, and end the confusion about why there is a speed limit once and for all?

A:

In everyday life, we experience the universe in a non-relativistic classical way. 
We are familiar with the concept of time and space. 
Defining the velocity as the ratio between a distance traveled in a given time interval is a much more natural choice instead of defining the rapidity.
If we all lived at relativistic speeds, or close to the event horizon of a black hole, or if we were small as an atom, we would use other tools to describe the universe around us. 
However, in our case, the quantities of classical mechanics work quite well and we can have a direct intuitive grasp of their meaning.

A:

It is not particularly easy to add rapidities pointing in different directions. 
e.g. Suppose B moves with rapidity $\rho_{1}$ with respect to A in the common $x$-direction of both of them. And suppose that C moves relative to B with rapidity $\rho_{2}$ in the common $y$-direction of B and C (we are taking it that their axes are aligned in this way). Now what is the rapidity of C relative to A?
You will not find it easy to answer.
The fact that this sort of thing is non-trivial is another reason (in addition to some good ones given in other answers) why it does not help to replace velocity with rapidity.

A:

People understand what velocity means, and how it is used.  Anyone who needs to use relativistic models knows enough to distinguish between which model they are using.  No additional words were needed when it was created,  and trying to invent new words now would just cause more confusion, not less.

