Define the formal structure of a Pushdown Automaton (PDA), including the stack mechanism and transition functions. Prove that a language is context-free if and only if it is recognized by a PDA. Provide a theoretical comparison between the computational power of Finite Automata and Pushdown Automata, specifically explaining why a PDA can recognize the language {a^n b^n | n >= 0} while a Finite Automaton cannot.