When you start or load a simulation, the random number generator is seeded. The way computers do random numbers, it actually produces a list of numbers that's always the same for the given seed. So yes, you could run two simulations, with identical seeds, on two computers, for years, and in theory they should always behave exactly the same even down to which mutations occur when.
If you don't want this behavior, try turninig off the random number seeding in the options panel, saving, and reloading the game. This will seed the generator with your computer's time, which should be different every time you run the program.