The simulation runs at 1200 cycles per second (depending on your computer and browser speed). That is a bit too slow to have the clock running exactly in the same way as on real hardware. The clock software has a prescaler that counts four edges on the 2Hz hardware signal, to get a one-second tick. The simulator has been tweaked to disable the prescaler, and the one-second-signal is simply generated after each multiplexing "frame" (so the counter is free-running and not coupled to real seconds). The memory locations that are used by the prescaler (0x058 and 0x0030) are manipulated by the simulator. The clock software itself is unchanged. Buttons for setting the time are not available in the simulator, mainly because the software would not scan them frequently enough.
In the files section, the software zipfile was updated with this new version.
Note that you can just change the software in the leftmost window, press assemble and run, and see how the program behaves ! For instance, it is easy to change the starting time of the clock from 00:00:00 to another value.