Pixel values over 255
There are image formats that use 2 bytes to represent value of each subpixel, that's values 0-65535 - but not for the reason you think. It doesn't make the max color brighter (65535 is exactly as bright as 255 was in 1-byte formats) but gives you more steps to work with.
In other words, 0 always means 0% brightness and max value (255 or 65535) always means 100% brightness. For max value of 255 the smallest color difference that can be represented is 0.39%, while for 65535 it's 0.0015%.
It doesn't make sense for images intended to be displayed as-is to use such precise color - human eye can't distinguish such subtle differences anyway - but it's useful for images intended for further processing. One example would be video game textures, which are affected by lighting etc. and with just 1-byte deep color channels processing could produce color banding.
Color gamut
Monitors dedicated for color sensitive work have wider color gamut, which means that they can produce a wider range of colors. On such screens the same color value of 255 (or 65535) looks more vivid than on a regular screen. If colors on an older/cheaper laptop screen look washed out compared to a newer/more expensive laptop, it's probably for the same reason: the more expensive one has wider gamut (although still not as wide as professional stuff), and usually worse contrast ratio too. 1366×768 screens were notorious for being relatively bad at this.
This could be what you had in mind when asking this question, except the input color values are the same. It's just that the max input value corresponds to a more vivid color, and the same increment in input values produces a more pronounced difference.
By the way, these professional monitors are too vivid to use them for normal stuff, like web browsing etc. - everything looks extremely vivid, especially the greens. They usually can be switched to a "regular" gamut to make the colors acceptable for non-color-sensitive work.
HDR
HDR stands for a bunch of different things in the digital work. Here I'm talking about "HDR displays", ie. displays that accept color deeper than 8 bits and offer higher-than-average peak brightness. Essentially displays that combine the two things described above.
Most such displays accept 10 bits of color per channel, resulting in a 0-1023 range. But if the display's peak brightness is very high, then 1023 doesn't correspond to the 100% of brightness that you're used to - it's more than that. If the image source (eg. computer) is aware of it, it can display "regular brightness" stuff without making use of the extra 2 bits of color (ie. maxing out at 255), but it can also use of the higher brightness in selected areas to achieve impressive dynamic range. So now when you're looking at the sun in a video game, the display will actually be blinding bright in that area, producing a more realistic effect.