We are learning a topological structure here. So mapping the neighbors in the lower dimension is the necessary and fundamental objective of SNE. Note that, in lower dimension we don't have much space to accommodate all the neighbors.
for motivation note that, we can accommodate maximum $n+1$ equidistant points in a $n$ dimensional space. So, what will a basic SNE algorithm do is collapse all the equidistant point to one point in lower dimension. This phenomenon is called Crowding probelm.
To mitigate this problem t-distribution was suggested. As it has a heavy tail it allows those points suffering from the crowding problem to be placed in a somewhat distant place (but not too much).