The 1060 6gb is light years ahead of the 2400g. I am not trying to knock the iGPU of the 2400g, it is very good for integrated graphics, but you are comparing a sports car vs a minivan. For gaming performance, nothing has a bigger impact than the GPU. So I suggest building your system around a GPU.
If you are not going to use the iGPU in the 2200g/2400g, then you are not using some of the more costly features. The 8100 is cheaper than the 2400g and will have better single threaded performance and single threaded performance is what the vast majority of games like.
You also have the bios update issues with the 2200g/2400g. Any 300 series AMD board will need a bios update to run those processors. In order to update the bios you will need a compatible CPU. AMD has a "loaner" program where they will ship CPU to update the bios, but is sounds like it could be a pain. When the 400 series chipsets are released it wont be an issue, but for now it is something to consider.
Prebuilt PC are an economical way to go right now, but they do cut corners on some parts such as the PSU, so you should be aware. This build is right at your budget and is a solid midrange system.
PCPartPicker part list:
https://pcpartpicker.com/list/yMD4Cb
Price breakdown by merchant: https://pcpartpicker.com/list/yMD4Cb/by_merchant/
CPU: Intel - Core i3-8100 3.6GHz Quad-Core Processor ($109.89 @ OutletPC)
Motherboard: Gigabyte - Z370P D3 ATX LGA1151 Motherboard ($95.89 @ OutletPC)
Memory: G.Skill - Ripjaws V Series 8GB (2 x 4GB) DDR4-3200 Memory ($95.99 @ Newegg)
Storage: Samsung - 850 EVO-Series 250GB 2.5" Solid State Drive ($99.99 @ Samsung)
Video Card: Gigabyte - GeForce GTX 1060 6GB 6GB WINDFORCE OC 6G Video Card ($354.98 @ Newegg)
Case: Corsair - SPEC-02 ATX Mid Tower Case ($49.99 @ Newegg)
Power Supply: SeaSonic - S12II 520W 80+ Bronze Certified ATX Power Supply ($29.99 @ Newegg)
Total: $836.72
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-03-09 09:33 EST-0500