I think there is a small mis-understanding in here among posters. I will try to explain the mis-understanding. Feel free to correct me.
We know a square display can be respresented by any NxN matrix. Each element within the matrix contains a number, a set of 3 numbers, representing the RGB value, or in another word, color. Since the number of columns is the same as the number of rows in the matrix, so therefore the ratio between the column and row is 1:1. In terms of display, people don't use the term "columns" and "rows", they simply say height and width.
Graphics are displayed by pixels, but graphics aren't stored as pixels, they are stored as a matrix defined above. To display graphics on the display, a function is needed to map each pixel on the display to the value of the graphics' matrix. Fewer pixels on the display means less details. The mapping functions used is also known as "Resolution".
Here comes the confusing part. More pixels doesn't mean higher resolution in some cases. For example, 3 tiny display, 1x20, 4x4, and 4x3. If the graphic is 4x4 big, then only the 2nd display can display the entire graphic without losing details, with the third can shrink the picture in a 3x3 matrix and the first have to shrink it to a 1x1 matrix. Although the first display has to most number of pixels.
In practical cases, graphics are far bigger than 4x3 or 16x9. However, movies are recorded in the ratio of 16:9, and therefore best being displayed as 16:9. There are 3 ways to display this on a 4:3 display. First, leave top and bottom will be blank and display it at the center portion. Second, resize it to 4:3. Third, cut out the left and the right portion. As to the quality, it depends on 2 factors, namely the way that is being stored, and the way it is being displayed. Better quality requires more space to store it, and therefore blueray disc movies are better quality than DVD, and DvD is better than CD. If the encoded(stored) quality is higher than what the display can show, then you lose some quality.
Gaming graphics are different from movies, as the display is a "viewport". The resolution represents the size of the viewport, that means, higher resolution means bigger viewport. The ratio becomes less important as all pixels are used, unless the game itself doesn't support it. 1024x768(XGA) shows the same viewport compare to 2048x1536(QXGA), that means the size of the world that got displayed are same, but less detail in XGA in comparison. Many games even support multi-display by displaying the viewport across the displays. Depending of the size of the game art, if the game is designed at a lower resolution than the display, then either arts will be upscaled (more blurry) or setting the viewport futher(zoom out). On the other hand, if the game is designed at a higher resolution than the display, then the arts will be shrinked(losing details).
User Interface isn't a part of the viewport, but something that is placed on top of the viewport, UIs block more portions of the viewport on lower reslution displays in comparison, and therefore, it is better to use higher resolution display as you can either fit more UIs on the display, or simply see more out of the viewport. Some games allow UI resizing, but it isn't dependent on the size of viewport, but the size of the display.
Therefore, when it comes to gaming, regardless of what the ratio is, the more pixel, the better. Regarless of whether it is wider, or simply bigger.
I didn't spell check, and my grammar suck anyways, so draw whatever conclusion you want.