We know that the effects produced by a moving body depend both on the speed at which it is moving and on its mass:
$$\mathbf{p} = m \mathbf{v}$$
Therefore it is useful, to evaluate this effect, to introduce the momentum vector $\mathbf{p}$. The kinetic energy is (in general of an object of mass $m$ moving with velocity $u$), $$\mathcal K=mu^2/2$$
How is it possible to express well in words the difference between kinetic energy and momentum?
Related that I don't like:
I believe that there are good reasons why physics uses formula to describe their finding. Thus, asking us to skip the formulas and still providing "good" definitions is difficult. I'm not able to do so, but I'll try explain my formulas and use them only to clarify what I mean.
On a high school level momentum and energy are related by the concept of force: Momentum is given by the sum of the force over "small" time intervals $$ p = \sum_i F_i \cdot \Delta t_i $$ where I use $\Delta p = p - p_0$ and assume $p_0=0$. By dropping the sum and considering a single time interval we obtain $p = F \cdot \Delta t$. Thus, momentum is the ability of a body to exert force over a "short" time interval.
In contrast energy is related to work, which is given by the sum of the force over "small" distance intervals $$ E = \sum_i F_i \cdot \Delta s_i $$ Again, we drop the sum, $E = F \cdot \Delta s$. Thus, energy is the ability of a body to exert a force over a "small" distance.
Since time and distance are rather different, the quantities momentum and energy are rather different. I believe that the bullet/rifle example described here is great. Nevertheless, here is my own example, utilising the concepts described above: Let's assume we like to drive a nail into a piece of wood. There are two equally valid perspectives:
Both perspectives are true. Therefore, the hammer acquires momentum and kinetic energy. These two concepts are used to answer different questions:
After you edited you question, I have to add this paragraph to my answer. Please do not change the focus of your question.
I understand that you are looking for simplified explanations and believe that this is the right way of teaching at high school level and during the first year of university. However, if one uses such simplifications one should ask oneself (a) how wrong is it, and (b) does it really help to understand the key concept. In my opinion the simplified explanation
the momentum gives us the effect that we observe when the sphere impacts on a surface (for example if a car impacts against a wall we observe fractures and visual damage)
is too vague to be helpful. Nobody knows what "the effect that we observe" means.
Kinetic energy is the energy possessed by a body in motion. This can easily be derived from Newton's Second Law as follows. $$\begin{align}dW&=F\;dx\\ dW&=ma\;dx\\ dW&=mv\,\frac {dv}{dx}\,dx\\ \int dW&=\int mv\,dv\\ W&=\frac 12mv^2\\ \text{Kinetic Energy}&=\frac 12mv^2\\ \end{align}$$
Momentum is a physical quantity that is possessed by a body in motion which is defined as the product of its mass and velocity. It is a vector quantity and has direction. $$\vec p=m\vec v$$
We can relate them mathematically by using the two formulae but intrinsically, they are separate from each other. You cannot derive one from another. A great example of this is how elastic collision between bodies takes place. Both kinetic energy and momentum are separately conserved.
