What difference do trans and cis configurations of amide groups bring to the polypeptide chain?
Apparently the trans configuration is more stable, so more amino acids
  with a trans configuration can be found within a polypeptide chain.
  Does this pose any significant characteristic to a protein, and if so
  what is this characteristic?


As you said, the trans- form is energetically more favourable. However, in proline the cis- form is energetically similar to the trans- form. The cis- structure introduces a kink or bend in the peptide chain which is seen in case of proline (see this section of wikipedia page on proline).