Practical applications of the palindromic substring problem?
Lots of string-related algorithms have applications in the field of bioinformatics. Motivations for finding palindromic substrings in bioinformatics include 


Finding secondary structures in RNA folding 1
Predicting DNA breakage during gene conversion 2
Finding/analyzing clustered regularly interspaced short palindromic repeats (CRISPRs) 3


These three motivations are presented in an article here and there are most likely many more.