Does more power really mean louder sound?

Power ratings on subwoofers are just thermal ratings that are sometimes over rated, and sometimes under rated.  But they will not tell you how much power the subwoofer will handle since that changes with the enclosure used.  With larger enclosures, it will take less power to reach Xmax.  Now to answer your above question, I will assume the same size enclosure is used.  Theoretically you would expect to see a 3 dB gain in output by doubling the power like that.  But other factors are present and a lot of the time it doesn't go up that high due to BL and power compression.

Power is getting cheaper and cheaper.  Because of this, many manufacturers are designing their subs to work in smaller and smaller enclosures.  This means that the subwoofers are more inefficient, but it is likely that the customer won't mind buying more power for the smaller enclosure.