The 5820k will require the x99 platform (basically everything costs 2-3x more) plus its a far more expensive processor. It also has a lower base clock and more cores. Gaming mostly wants fast as possible single thread execution speed not more cores, so real advantage there.

The 5820k has 2 main 'advantages' over the 6600k. First it uses quad channel DDR4 ram vs the 6600k's dual channel, but for real world gaming this makes little to no noticeable difference.

2nd, the 5820k has 28 pci-e lanes, the 6600k has 16 pci-e lanes. If you are only going to install 1 GPU in the system and nothing else card wise this doesn't matter.

If you say install a GPU and a sound card then it will have to split up those 16 lanes running the GPU at 8 lanes instead of the full 16 and using the spare ones for the sound card. This is fine because modern GPU's cant cap 8x pci-e 3.0 YET, but in the future they will be able to. So if you get a new GPU a few years down the road you -might- find it bottlenecked. If all you are going to do is run 1 GPU in the system and nothing else however you don't need to worry about that at all however.

Unless you are just building a extreme rig for the hell of it cuz you have the money to blow or are doing heavy workstation related tasks I would just get the 6600k, its going to be cheaper and probably actually perform better for gaming provided you are just going to be using a single GPU setup.