It depends, but probably not.
The main difference is the 780i has PCI Express 2.0, and I think the 680i is only 1.0. That might make a difference with some of the newer graphics cards, but it sounds like you've got a somewhat older system with a somewhat older card, so it probably won't matter much unless you also replace the video card.
Also, the 780i boards support the newer 45nm quad-core chips, whereas some of the 680is do and some don't. Only an issue if you want to upgrade your CPU.
Basically, even if you do get a small performance change, it's going to be insignificant compared to the difference you'd see from upgrading the CPU and video card. So unless you're planning on doing that also, if you ask me, it's not worth the trouble of installing a new motherboard and reinstalling your OS in the process.