You may enjoy the book "Understanding Exposure" by Bryan Peterson. It pretty much starts out by explaining exactly what you've discovered... but takes it a step further. As Overread explains... as you change the exposure settings to keep the needle centered, the equivalent exposures actually do result in images which look different BECAUSE you are changing things such as the depth of field.
For any given shot you are about to take, there are many many combinations of shutter speed, aperture, and ISO which will achieve the same exposure on the sensor... but ONE of those combinations will look better than the rest.
Now that you understand how you can play the exposure triangle trading game... the next step is to learn WHY you'd want to play the trading game to get a better looking result.
incidentally... there is a 4th factor that never seems to get mentioned. Do not forget that you CAN actually change the amount of light falling on your subject... either by adding your own light via flash or reflectors... or by simply MOVING your subject to a location with better lighting (and better background.)