AMD vs Intel - Which to choose?

For your use either will work just fine!

The requirements you've specified doesn't need a lot of CPU power. An AMD 5800k APU is the best choice for you, if you want a good balanced system that can handle light gaming as well.
If your looking for raw CPU power, get an i3, but you'll need to buy a discrete graphics card separately, and that might exceed your $500 budget.

I'll give you a rough skeleton to start with, you can fine tune it the way you want to,

AMD A10-5800K APU
Any cheap FM2 board (buy only a ASUS or Gigabyte board)
4GB DDR3 1600MHz RAM (faster the speed, the better, see if you can get 1866 rated RAM for cheaper)
500GB WD Caviar green HDD

keyboard, mouse , cabinet and monitor is totally upto you  

i3/i5/i7 includes integrated graphics as well, no video card needed unless there is a going to be some gaming going on.

Key question: do you live near a Mirocenter?  
http://www.microcenter.com/

Also +1 to festerovic for the Dark Tower quote...

Eh. Phenom II x4 965 is £40-£50 cheaper than an i3 2100. £50 will get you a 6670 DDR3.

If you do get i3 get the one with HD 4000 no reason to have less igp. It's better for movies than Hd 2500 or 3000.

Otherwise an APU is good for you I agree. It can play 1080p movies fine and the boards can come with HDMI outputs that it can actually utilize fully. It has better graphics processing power in the CPU than any of Intel's CPUs as of today.

For movie playback, it makes no difference. Any SB/IB IGP will be capable of playing online streaming and BD movies. Even without hardware assist, the i3 would still have enough processing power to handle full software decode.

The i3-3225 costs ~$10 extra and is not really worth it unless planning to do some low-end gaming on a GPU-less PC.