I know this has been explained many times before in various places, but once again, here is a brief explanation of the way "widescreen" works with 480i/480p signals...
Unlike 720p and 1080i signals, which are widescreen by nature, 480i and 480p signals are not inherently widescreen. To output a widescreen signal in these resolutions, source devices send out an image which uses non-square pixels. In such images, the pixels are taller than they are wide, i.e. "squeezed" horizontally. If viewed exactly "as-is", everything will appear "too skinny" or "too tall". This is known as an "anamorphic" video signal.
Since they're analog signals, they don't have a formally defined horizontal resolution, but it's enough to note that the resolution does not change, regardless of whether the source device is sending out a non-anamorphic (with roughly square pixels) or an anamorphic (with "skinny" pixels) image. The amount of resolution is fixed.
If a device always sends out an anamorphic signal when configured for 16x9 mode, then it's a trivial matter to set the widescreen TV to its anamorphic mode ("Fill" or "Full") and leave it alone. However, many devices selectively send out an anamorphic signal, i.e. change what they are sending based on source material. In some cases, the devices are able to signal at power-on that they are broadcasting anamorphic content, causing the TV to automatically switch to anamorphic mode. However, some devices, such as the Wii, selectively send out an anamorphic or non-anamorphic image without any additional signalling. As such, the user must *manually* change the TV between 4x3 and 16x9 modes if all content is to be watched in its original aspect ratio.
This leads to these combinations:
1) TV is in 4x3 mode and Wii is broadcasting non-anamorphic content. Image has correct aspect ratio, and black bars (or grey bars) on the sides (generated by the TV).
2) TV is in 16x9 mode, and Wii is broadcasting anamorphic content. Image has correct aspect ratio, and fills the entire screen.
3) TV is in 4x3 mode, and Wii is broadcasting anamorphic content. Image appears too skinny, or horizontally compressed.
4) TV is in 16x9 mode, and Wii is broadcasting non-anamorphic content. Image appears too fat, or horizontally stretched.
If Nintendo were wise, they would make the Wii always output an anamorphic signal when configured for 16x9, i.e. squeeze even the 4x3 content. This would make it painless for users to leave their TVs in 16x9 mode, but... alas.