I know that the cristae of the inner membrane can be tubular or vesicular but I would like to know what are the functions of each .
There is a reasonable speculation. That is to get more areas where ATP synthases are localized in order to produce ATP. But this doesn't tell why the structure has to be the exactly that shape.
As the review Ilan show, the shape seems to be determined by ATP synthase dimerizetion. The shap may be just a consequence of ATP synthase dimerizetion, although it successfully enlarges the area where ATP synthase is localized.
The general morphology of cristae is dictated by ATP-synthase Complex dimers, which form rows along the edge of cristae, sort of pinching the inner mitochondrial membrane together (see http://www.pnas.org/content/108/34/14121.short). The angle at which the dimers form, which is different in various species, causes different cristae morphologies. In organisms that don't have ATP-synthase or in yeast strains that are devoid of key subunits required for dimer formation, the cristae of their mitochondria are much more irregular and have a less defined shape, often described as 'onion-like' in the literature (see http://www.jbc.org/content/279/39/40392.short).
Therefore, I guess you could argue that actually it's the form that is a product of the function, rather than the other way round i.e. different cristae morphologies having different functions. The purpose, or selective advantage, of mitochondria having cristae is most likely that they create a greater surface area that can facilitate a larger number of complexes of the electron transport chain, which would enable greater ATP production per organelle. This explains why in eukaryotes that don't produce ATP via the electron transport chain there is no selection to maintain cristae in their mitochondria, or homologous organelles such as hydrogenosomes and mitosomes, and so usually the cristae are lost altogether in such species.
Other protein complexes are also involved in cristae formation and general architecture. Search for MICOS and MIB complexes to find out more.