  We make a theoretical attempt to compare the predictions from normal and
inverted hierarchical models, within the framework of $\mu-\tau$ symmetry. We
consider three major theoretical issues in a self consistent ways, viz.,
predictions on neutrino mass and mixing parameters, stability under RG analysis
in MSSM, and baryogenesis through leptogenesis. We further extend our earlier
works on parametrisation of neutrino mass matrices using only two parameters,
in addition to an overall mass scale $m_0$, and the ratio of these two
parameters fixes the value of solar mixing angle. Such parametrisation though
phenomenological gives a firm handle on the analysis of the mass matrices and
predicts lower values of solar mixing angle in the range
$\tan^2\theta_{12}=0.50-0.35$.We check the stability of the model under
radiative corrections in MSSM for large $\tan\beta\sim 58-60$ region and
observe that the evolution of $\bigtriangleup m^2_{21}$ with energy scale, is
highly dependent on the input high scale value of solar mixing angle. Solar
angle predicted by tri-bimaximal mixings and also values lower than this, do
not lead to stability of the model. Similarly, the evolution of the atmospheric
mixing angle with energy scale at large $\tan\beta$ values, shows sharp
decrease for the case with non-zero value of m_3. We apply these mass matrices
to estimate the baryon asymmetry of the universe in a self consistent way and
find that normal hierarchical model leads to the best result. Considering all
these three pieces of theoretical investigations, we may conclude that normal
hierarchical model is more favourable in nature.
