Explanation :
Since all the edge weights are unique, hence the minimum spannig tree of the graph will be unique.
Shortest path betweeen the two vertices need not to be unique. A counter example for the statement can be,
The path from $ {\mathrm V}_1\rightarrow{\mathrm V}_3 $ can be,
I. $ {\mathrm V}_1\rightarrow{\mathrm V}_2\rightarrow{\mathrm V}_3\;:\;1+2=3 $
II. $ {\mathrm V}_1\rightarrow{\mathrm V}_3\;:\;3 $
Hence the path is not unique.