What is the actual reason for geometrical isomerism?
Geometric isomerism occurs when two structures with the same connectivity are not interconvertible.

Cis-Trans isomerism is common and easy to recognize kind of geometric isomerism. The carbon-carbon double truly has limited rotation. The double bond in the alkene functional group consists of a $\sigma$-bond located within the plane of the molecule and a $\pi$-bond located perpendicular to the plane of the molecule. In the image below (created by David Pilz for Wikipedia), you can see the $\pi$-bond in a perpendicular plane.



The $\pi$-bond is made from $p$-orbitals, so it cannot have the same symmetry as the $\sigma$-bond. You can see the formation of the $\pi$-bond from $p$-orbitals in this picture by Wikipedia user JoJan. In order for the $\pi$-bond to exist the $p$-orbitals must be aligned. 



In order for the molecule to rotate from $cis$ to $trans$, the bond axis would need to rotate so that the $p$-orbitals are no longer aligned and can no longer form the $\pi$-bond.  This slide show from James Condon at Roane State(PDF) explains the same concept. Rotation around the double bond requires enough energy input to break the $\pi$-bond, at least $250 \ \text{kJ/mol}$, which is a lot for room temperature.

The clearest demonstration that geometric isomers are clearly different molecules is to look at some examples with drastically different properties. It is important to remember that the experimental observation that these compounds with very similar structures have very different properties predates the explanation that the double bond has limited rotation.

The two isomers of 1,2-dichloroethene have different physical properties due both the difference in polarity and the difference in shape:


  $cis$-1,2-dichloroethane





Dipole moment = $1.9 \ \text{D}$
Boiling point = $60.2 \ ^\circ\text{C}$
Melting point = $-81.47 \ ^\circ\text{C}$
Density = $1.28 \ \text{g/cm}^3$



  $trans$-1,2-dichloroethane





Dipole moment = $0.0 \ \text{D}$
Boiling point = $48.5 \ ^\circ\text{C}$
Melting point = $-49.44 \ ^\circ\text{C}$
Density = $1.26 \ \text{g/cm}^3$



  However, my favorites are maleic acid and fumaric acid. Notice the extreme difference in water solubility as well as the difference in acidity. These molecules behave very different chemically.


Maleic Acid $cis$




Melting point = $135 \ ^\circ\text{C}$ (decomposes)
Density = $1.59 \ \text{g/cm}^3$
Solubility in water $788 \ \text{g/L}$
Acidity $\text{p}K_{a_1} = 1.9$, $\text{p}K_{a_2} = 6.07$


Fumaric Acid $trans$




Melting point = $287 \ ^\circ\text{C}$ (does not decompose)
Density = $1.635 \ \text{g/cm}^3$
Solubility in water $6.3 \ \text{g/L}$
Acidity $\text{p}K_{a_1} = 3.03$, $\text{p}K_{a_2} = 4.44$