Real life applications of finite automata
As Yuval Filmus said, the KMP substring matching is widely used in practice because of its modularity and efficiency.

A maybe more "practical" exemple is lexical analysis.
(https://en.wikipedia.org/wiki/Lexical_analysis) 

Depending on your application (let say or the compilation of a programming language) the parsing phase can be done with some various 
generalisation of automatons (state transition machines or pile-automatons for instance). The language they accept is infinite (think of the set of all programs that are syntactically correct in your favorite language), yet
the recognition of a correct program should be done by an automata (or a generalisation of it) which is finitely presented.