When GP4 came out it hammered almost any PC of the period and even today it requires fairly decent hardware to get the most out of it.
The quad core CPU will help, but the slower clock speed will detract from that a little. Both CPUs are barely adequate really. However the GFX card should give a noticeable improvement over your old one.
Nearest equivalent I had in the past (E5430, GT720, 4gb RAM) ran GP4 at about 25fps with 2gb RAM and about 30fps with 4gb, but you'll need to try a few f1graphics.cfgs files to get the most from it (plenty on the forum - give several a try to find the best one for you). Also, depending on the RAM you have, install the 2gb or 4gb patch - it always seemed to give me a boost.
Others will recommend you don't let GPXPatch use more than one core. They might be right, but I always use all of them - and you'll probably need to as well if you want to have a playable game.
That said, if you don't really need Gate's kludgeware, Linux is definitely a great choice for GP4.