Example: The police arrived and the police chief says: I want to know if there is anybody in the house at the moment!! or..at this moment?
I think the main difference is emphasis.
In the example sentence, exchanging 'at the moment' for 'at this moment' will shift the emphasis to the time, 'THIS moment', making it a more important part of the message that he wants to know if there's someone in right now. I'd actually rephrase the sentence to 'right now!' instead of 'at the/this moment!', it gives more energy to the message in my opinion.