I talked about this on another forum I visit.
I took the two monitors I have attached to my computer and took one screenshot of each.
One is a widescreen monitor, and one is not (Though to be technical, one is 16:10 and the other 5:4, so they aren't the exact aspect ratios Brawl uses, but they are close enough to use as an example).
Over all, you won't really benefit more from one or the other but I do think that widescreen has an advantage when it comes to games with obnoxious HUDs. Such as the WoW interface, and the Smash Bros damage percentages.
For example, I loaded WoW up on my 1280 x 1024 monitor and took this screenshot.
That is certainly playable, but I feel like the quest log and the game chat are kinda in the way.
Then, I loaded it up on my 1920 x 1200 monitor and took this shot.
They're still in the way, but now I have much more space around my character for the environment and other characters.
If you look at the top of the non-widescreen screenshot, you'll notice that you can see more of the brown 'destroyed tent thing' hanging down than you can in the widescreen one and you can see more of the stone building on the right side of the screen in the widescreen one.
Same with the Smash screenshots. Notice how in the widescreen one, sonic has less visible above his head, and in the standard one, there is less visible on the sides of the image.
So, the only real advantage you'd have on a Widescreen display would be the damage percentages are spaced out a little bit.
I'd say that the bottom line is this:
If you have a widescreen HDTV, run it in 16:9 mode.
If you have a standard Television, run it in 4:3 mode.
Now, I know someone who has an old HDTV that is 4:3 that has a widescreen 'mode' so that they can watch their HD channels.
I'd have to say that if you are thinking about putting borders on a screen like that by entering this mode and then setting the system to 16:9 mode, you would probably be better off just running the game in 4:3 mode instead.
In my opinion, the advantages of having the percentages spaced out aren't worth the screen space that would be wasted by the black bars.