I agree with Rohan and Madhumita.

In other words, Gene signature is a list of genes whose expression is correlated with a biological state of interest (example: down gene signatures).

Also it reveals the connections between drugs and diseases (new therapeutic opportunities) like predicting survival after chemotherapy.

Biomarker is a characteristic that is objectively measured and evaluated as an indicator of normal biological processes, pathogenic processes or pharmacological responses to a therapeutic intervention.

examples: Troponin T is a biomarker for AMI

alpa- feto protein is a biomarker for liver cancer

the above markers are approximately absent in normal state, and elevated in disease. Some biomarkers can be used to monitor therapy.

A biomarker is a molecule that can be used as an indicator of a particular biological state. For example, It could be used in diagnosis, prognosis of disease evolution or prediction of therapeutic response. Recently, with the development of highthrouput techniques such as microarrays, was posible to study the relationship between  a group of gene and biological states.  Since then a group of gene could be used as a biomarker and is named gene signature.  

The difference lies precisely in the concept of signature. Which is a specific and complex combination of biomarkers that reflect a biological state. They are much more precise and specific that patterns of biomarkers or single biomarkers from a particular state. Sometimes, the complexity of the set of biomarkers that make the firm goes beyond the simpel presence and concentration of the markers but involves the kinetics of these.