I'm building a new computer with an HD 5850. I was wondering if my current monitor would give me a full HD experience, or if I should upgrade to a monitor with 1920 x 1080. Would there be a noticeable difference? Currently, I'm using a 20" monitor on 1680 x 1050.
There will be a difference. Think about it, your missing out on 240 horizontal pixels and 30 vertical pixels. When you multiply those numbers together that is alot of area that you will not have.
Also, true HD 1080P has the dimensions of well..1080. You won't be getting the best bang for your buck in terms of your GPU if you don't upgrade to a monitor that is capable of showing 1920 x 1080. They are fairly cheap too. Hope this helps.
There's nothing particularly magical or special about 1080p. Unless you're specifically watching 1080p video, there's no specific reason to target 1080p any more than any other resolution.
That having been said, a higher resolution will always look better. A 20" 1680x1050 will look nice, a 24" 1920x1200 will look nicer, and a 30" 2560x1600 will look nicer still. There are other factors to consider too. Image quality varies widely between monitor types, with most cheap LCDs sold having TN panels (lowest quality), and the higher end models having VA and IPS panels (which look much nicer IMHO).
If you're forking out for a HD5850, its kind of overkill if you're sticking to 1680x1050 resolution you might as well save yourself £100 and get a cheaper graphics card like the GTX260. If you want to see the performance difference of the HD5850 compared to last years cards then I would suggest getting a monitor with at least 1920x1200. Will be excellent for gaming.
Personally I would get the Samsung SM2433BW 24". You won't regret it.