There exists examples of different  point configurations in $\mathbb{R}^2$ having the same the set (but different matrices of!) 
 distances. The simplest example contains 4 points and could be found in the paper of Boutin and Kemper, see http://arxiv.org/pdf/math/0304192v1.pdf  -- scroll to page 5 to see the picture
[*Added by J.O'Rourke*]:
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![Fig4][1]

It is shown though (also Kemper, I believe) that for most configurations the set of distances determine the configuration (which is probably intuitively expected). 

The example I have mentioned answers your question, but actually it would be natural if in your question you also require that the distances come with their multiplicities. 


  [1]: https://i.sstatic.net/5zXDB.png