When a device is supposed to have a "separate ground" I believe it means that the ground is supposed to be "quiet". In other words, the ground voltage should not change very much. You might ask, "what would cause the ground voltage to change?" The answer, "current will cause the voltage to change".
So, if you have a microprocessor connected to ground (0V) and some other thing connected to the same ground that is dumping lots of current, then maybe your ground isn't really 0V like you would have hoped (V=IR). In this case, you need to make sure to hook things up in a way to minimize ground moving around. In other words, you need to "separate the grounds."