Why must the particles of an ideal gas be point-like? 
Why is a gas of elastically colliding hard balls of finite size not
  ideal?

Respectively: 

Why is it essential that the particles of an ideal gas are point-like?

Especially:

Which differing thermodynamic properties of a gas of not point-like particles are most striking in comparison with an ideal gas
  (of point-like particles)?

 A: Because if those particles aren't point objects, you must also take into consideration that they take some space in the system and have properties like density and size which have to be taken into consideration when formulating the laws. This makes everything extremely complicated. A huge part of classical mechanics is only true for point objects for the very same reason.
A: There are multiple ways of looking at the ideal gas model. One is to say we have point particles colliding elastically (as an ideal scenario) and proceed to obtain the exact equation of state, i.e $pV=nRT$. The other approach, is to state a priori the relevant length scales and time scales at which the system is going to be studied are sufficiently macroscopic, so as to be able to disregard any microscopic structure in the constituent particles. In both cases, the excluded volume effect (which is what the effect of existence of a finite size of particles in a gas is called) is neglected, as a model assumption in the first case and as a practical approximation in the second. The simplest fluid models that can be solved analytically are the so-called hard sphere models which involve the crudest form of binary excluded volume interactions. In such a case, one can in principle write down the entire virial equation for pressure (the virial coefficients being calculable from microscopic parameters of the model, size of the particles included) and hence the equation of state for this fluid is derivable. The Van der Waals equation is a specific mean field limit of such a virial expansion. The effect of excluded volume will be most prominent a low temperatures and high densities and it will have a substantial effect on the condensation of the fluid (the critical temperature and the scaling laws). 
A: If the particles are not point-like, they will take up some volume.  As the gas is compressed, the collision frequency will rise more quickly, which will make the pressure-volume curve change.  The corrections in the Van der Waals model of a real gas account for the volume of the particles.  Also if they have internal structure, that structure can have degrees of freedom that change the specific heat.  The fact that nitrogen and oxygen are diatomic leads to the specific heat of air being $\frac 52R$ at normal conditions.  There are two rotational degrees of freedom available along with translation.
