Definitely go for 64-bit; you need its address space capabilities to handle the memory on your RAM, video card, etc. If you go with 32-bit, you'll have less than 2GB of your main system RAM available. (That 5970 has 2GB onboard!)
Go for 64bits. You have 4GB of RAM and a processor that support 64bit. There is no reason to stick with 32bit.
Almost every 32bit-only applications will run fine in your system because of the WOW64. Sometimes It will run slower or even faster than a 32bit OS (Like, 3-10% difference).
For any system capable of running 64-bit (and has 4 GBs of RAM or more), I'd recommend going with 64-bit. Most programs designed for the 32-bit version of Windows will work on the 64-bit version of Windows. Whereas programs designed for 64-bit Windows will not work on 32-bit Windows.