The second death is what the eternal punishment in the pool of fire is called, they are not two different things. Your real question is more like: "why is the second death eternal punishment in the pool of fire and not the complete annihilation or temporal punishment of the wicked?".
Why? Because quite simply it is what the Lord, in his most holy and righteous judgement, has ordained to happen to those who depart this life in a state of original or mortal sin. It is what through the sources of revelation he has communicated to us. That those at his left hand, not found in the book of life, are thrown into the everlasting fire.
This is what The Catholic Church, following the mandate of her Lord believes and teaches. The belief in the eternal punishment of hell is, in fact, a De Fide dogma of the faith. Meaning that it is an article to which all those professing the Catholic faith adhere to.
Evo