Please explain the difference in thse cards compared to my system

The 6770 is a rebranded HD 5770, which is from the AMD's HD 5000 series. The 7750 is based on the newer (current generation) technology from AMD. The HD 7750 is NOT backwards compatible with PCi-E 1.0 on your motherboard with the first generation Socket 939 Athlon 64 X2. This is not due to PCi-E specifications but due to the way the cards are designed to draw less power. I have heard of several cases of HD 7000 series not backwards compatible with PCi-E 1.0. The 6770 *should* be compatible with your motherboard, although I can't guarantee that, because every GPU board maker (in your case XFx) may or may not guarantee backwards compatibility despite what the GPU chip maker (AMD in your case) says.

For the record, I have used the HD 4000 series on a system nearly identical to yours, and it was perfectly compatible. I would venture to guess that MOST HD 5000/6000 series should run fine too.

If you want to try a Nvidia, the equivalent to the HD 7750 is the GTX 650. That has a better chance to work on your older PCI-e 1.0 board, because Nvidia supposedly sacrifices power draw optimizations for back-wards compatibility.

Compatibility issues of PCI-e 3.0 cards with older gen PCI-e boards summarized here:

http://www.overclock.net/a/the-final-answer-to-the-controversial-pcie-x16-version-compatibility

Most relevant section quoted:

"
While it is true that PCI SIG while developing PCI express slots made sure that they remain compatible with each other, it is the GRAPHIC CARD manufactures (i.e Nvidia & AMD) and the sub vendors (Saffire, Zotac, EVGA etc) that really make the difference!
We all know that the AMD Radeon HD 7000 series is extremely power efficient. Having said that it comes at a cost. This power efficiency comes due to the new electrical interface featured in PCIe v3.0 and 2.0 slots and is ABSENT in PCIe v1.1 slots. Hence they are NOT COMPATIBLE.
Sources of proof: Online interaction with Sapphire technical staff
Online interaction with AMD technical staff
Personal interaction with technical staff from MSI
They all clearly stated that that PCIe v3.0 cards are not compatible with v1.1
Further to prove their point MSI was even kind enough to provide a reference card they had with them, i.e the HD 7850 and it did not work on my computer's motherboard featuring PCIe x16 v1.1 slot. Just to make sure I installed the card correctly I placed another card from Nvidia i.e GTX 560 onto the board that I got from a friend and everything worked flawlessly."

All the cards will be severely bottlenecked by your CPU and motherboard. You could get the 7750 over the 6770, the difference being the former is newer, therefore more power efficient, as well as being a more powerful.

What I recommend is getting the Intel Core i3 3220 ($120), Gigabyte B75M ($80) and 4GB RAM ($20). This equates to $220. You can then get the AMD Radeon HD 7770 for $100.

If this is out of reach however, then the 7750 is probably a good choice. Every other component is fine.

P.S. don't get the GTX 280: old, hungry and hot.

Yeah, that'd be a good idea. You can then always return it if it doesn't work. That's when I'd be looking into a new computer altogether.

P.S. Please choose a solution. Thank you.