What is the difference between finite automata and Büchi automata?
There is no difference in the specification (syntax). The only difference is in the interpretation (semantics). The language accepted by a DFA consists of finite strings defined in a certain way given the DFA, and the language accepted by the corresponding B&uuml;chi automaton consists of infinite strings defined in a certain way.

Indeed, you can leverage this connection to prove results such as:


  An $\omega$-language is recognizable by a deterministic Büchi automaton iff it is the limit language of some regular language.


For a proof, see Wikipedia.