This is definitely the ram. This is likely not the CPU, as it has 4 cores, but then again, it has no threads. 4GB of ram, especially DDR2, is not a lot for most games these days. These kinds of games stutter because if you drive really fast, it is trying to load all these different parts of the map, and 4GB just isn't enough to store all of the data. 6GB could be enough, you would see an improvement.
As for your new build, I would recommend this. (http://amzn.to/2tQmsyn)
It is a cheap, but good motherboard. It has a LGA1151 socket, which is better than the LGA775 socket you have, and compatibility to DDR4.
And a lot of people use a Pentium G4400 for gaming on a budget.
(http://amzn.to/2tqh0mt) which is dual core. You would be surprised on how good the processor is when it only has 2 cores. But if you want a quad core cpu (which is reasonable why), you could get something like an i5. The Pentium I linked is probably the best cheapest that you could get on this socket. And yes, it is known to have stuttering issues even with 8GB on GTA V, but the processor I linked or anything better than that should make it better.
Now, DDR4 ram. I would focus mainly on the CPU and the ram. The ram is kind of expensive, but for a 8GB kit (2 4GB sticks), it is around 51 euros, or 60 USD. (http://amzn.to/2vwtn1X)
I hope you the best of luck!