Asus is the leading brand in motherboards and top notch quality but Asrock have decent sub $200 mobos.
The p8z77-v pro is an ATX motherboard and has the full 7 expansoin slots.(3 graphics slots)It has 4 DIMM(ram) slots,however...
The Asrock mobo only has 3 DIMM slots and is a M-ATX(micro atx)board Wich has only 4 expansion slots.Overall the Asus mobo has more slots and is larger.
Both have Intel Z77 chipsets. Yes your are right.
Asrock was a side branch that had slightly been part of asus, and now has more or less became its own brand. as for your question:
Quality: Asus has highest quality, But Asrock puts more features on their boards for less than asus does(as asus boards have a premium cost)
Compatibility: compatibility is all depending on the motherboard model itself rather than the brand. compatibility issues are usually fixed with bios updates if necessary.
Reliability: both can make fine sub 200$ motherboards. the extreme4 series(including m albeit its micro so theres less room inbetween pci-e slots) has the best potential for its price range, Asus is just more reliable hardware as they have alot of experience with hardware and is a large name brand.