If the products are identical as you said, then, even if the other developer is building at faster speed than you are I would say you still have the upper hand because of your greater visibility.

That said, if I were you I would launch a beta, even if its only a private beta, just to get the word out there, that way you can preempt his launch by getting the word out there. See every feature that its not completely necessary to the minimal viable product, cut those and launch.

Bests of luck!