I would go with the Gigabyte model.
The Gigabyte model has heatsink on the VRM , the circuit which powers the cpu by converting 12v from the power supply to 1v..1.4v used by the CPU... this means the VRM will be cooled better and in general have lower temperatures.
Now, with a 4 core processor like 2400g it's not that important, because the CPU doesn't consume so much power that the VRM would be hot, but if you'll want to upgrade the cpu in the future to a six core or an 8 core, having heatsinks on the VRM will help.
So having heatsink on the VRM is a plus and makes the board better suited for overclocking. BOTH motherboards can do overclocking, but the Gigabyte model seems a tiny bit better.
A part of the VRM which powers the integrated graphics does not have heatsinks on both motherboards, so both motherboards are equally bad... you should not overclock the integrated graphics by a significant amount.
Besides this, the Gigabyte model has an additional pci-e x1 slot that's positioned well at the bottom of the motherboard and isn't blocked by a big video card you'd put in the pci-e x16 slot. The Asrock model has only one pci-e x1 slot above the video card slot.
So you have 2 pci-e x1 slots on the Gigabyte board and only 1 pci-e x1 slots on the Asrock model.
So that's all when it comes to hardware differences, the audio chip, the network chip are the same, the memory slots are the same amount (2), you have the same number of USB ports, SATA ports, same number of video connectors.
If you ignore the lack of a pci-e x1 slot, the Asrock is slightly better in the cable management department : the SATA connectors are right angle which helps route the SATA cables away from the board, instead of sticking straight up. Also, the USB 2.0 and USB 3.0 headers are in a slightly better position, near the center of the motherboard so you don't have to pull the cables from the front of the case all the way to the bottom of the motherboard.
Last, visuals... the Gigabyte is all black/gray while the Asrock one is a mix of black and white and lots of stripes... I like the all black more.
Oh.... and the Gigabyte has TWO ps/2 ports, which would allow you to use a PS/2 keyboard AND a PS/2 mouse while the Asrock has a mixed PS/2 keyboard/mouse port. But in real world... few people use ps/2 mice anymore so it's not that important.
And... the Asrock model has a COM header, which would allow you to add a serial port connector on a bracket but that's only relevant to you if you deal with connecting to devices using null modem cables and so on ... a usb to serial adapter is a couple of dollars if you ever need one, so it's not really such a big deal that it's missing from Gigabyte boards.
