The salamander is the symbol that is on the firemen's uniforms. It represents everything that they do, burning the books in the name of maintaining control of their civilization. It stood for power, fear, destruction of life and home, and burning as a lifestyle and career. It is a symbol of how burning completely and totally destroys, leaving nothing behind. The Phoenix is also a symbol of burning, but, it is a more hopeful symbol; one that represents the hope that comes from starting anew, and having a second chance. With the salamander, there wasn't a second chance; if the salamander arrived at your door (the fireman), then you were out of luck. But, the Phoenix is the creature, or bird, that rises out of the ashes of destruction and gives mankind a chance to start over. It is the symbol of hope, for Montag's society to rebuild itself and try to start again without making the same mistakes. So, the main difference between the Salamander and the Phoenix is that one represents burning without any hope or redemption, and the other represents burning with hope and redemption afterwards.
I hope that those explanations help a bit; good luck!