Why are allocation and distribution important consequences of the second welfare theorem?
The First Welfare Theorem essentially says


  price equilibrium with transfers => Pareto optimal


The FWT hence solves the "problem of efficiency" i.e. If we let the market work through prices, we will achieve Pareto efficiency.

The SWT, on the other hand, states


  Pareto optimal => price quasi-equilibrium with transfers* 


It assures us that any Pareto optimal point can be achieved via the market. 

Note that distribution and inequality are normative problems which must be addressed through say, the political process, rather than economics. The concept of Pareto efficiency does not tell us anything about distribution. Oftentimes in a 2x2 model, giving Agent 1 all the resources and Agent 2 none of the resources counts as Pareto optimal.

Therefore the SWT addresses the "problem of distribution" by saying that "Ok, society has decided that a certain Pareto optimum is the most desirable outcome. We know that we can achieve this by redistributing endowments/incomes and then letting the market work."

*PQWT is a weaker definition of PEWT