Shortest paths in weighted graphs, and minimum spanning trees
Let's consider the hypothesis of the question to try clarify every aspect:  

Our Graph G should:


be undirected
be weighted
be connected
with no negative weights
with all weights distinct
in this graph, the shortest path between any two vertices is on the
minimum spanning tree (MST)


I believe the key point here is the article used in point 6, as pointed out by David Richerby comment 

From the wikipedia page: Article (grammar):  

Definite Article  


  A definite article indicates that its noun is a particular one (or
  ones) identifiable to the listener. It may be something that the
  speaker has already mentioned, or it may be something uniquely
  specified. The definite article in English, for both singular and
  plural nouns, is the.


Indefinite article  


  An indefinite article indicates that its noun is not a particular one
  (or ones) identifiable to the listener. It may be something that the
  speaker is mentioning for the first time, or its precise identity may
  be irrelevant or hypothetical, or the speaker may be making a general
  statement about any such thing.


So point 6 lends to some ambiguity:  


It could be that the shortest path is implied to be unique. (In
this case I must say the hypothesis and the question are sort of
misleading)
Or it could be that the shortest path is only one of the
possible shortest paths between any two vertices


So if we interpret the hypothesis as per point 1 (i.e. the shortest path means there is a unique shortest path) then the hypothesis is also the answer.

If we interpret the hypothesis as per point 2 then we could consider the following undirected graph:

A--5--B
|   /
2  3
| /
 C


There is no unique shortest path between A and B
Since we could have both:
AB (cost 5)
AC + CB (cost 2+3 = 5)

So if we interpret the hypothesis not meaning the shortest path is unique and asks you to verify this, then we have demonstrated that there is such graph with no unique shortest path that satisfies all other conditions.