It's for the "Cable Select" feature.
The cable sets the device on grey connector as slave, and the black as master. Blue is for the motherboard.
The IDE devices have to be jumpered for "Cable Select" before the cable can set it.

To be short - all 3 80-pin cable connectors are different.
The blue connector (motherboard) has 1 pin (#34, wire #68) grounded and the wire not connected/cut between blue and gray connectors. So motherboards can detect 80-pin cables. Gray and black connectors have the pin #34 connected. If you use this cable "topdown" - it will be detected as a 40-pin one and will work as a 40-pin one.
There are cables with the un-cutted wire #68 (so it can work topdown as an 80-pin cable) but IIRC the ATA stardard wants wire #68 cutted.
The pin #28 (the wire #56) is used for the cable select option. The slave connector (the middle one, gray) isn't connected to this wire. Blue and black (master) connectors are connected.
And the pin #20 is physically omitted in all 3 connectors.

Thanks for the answers. One more question.

I noticed that the Asus ribbon cable connector is a bit narrower than the other ones I have usually seen but the pin holes are at same locations. I have not usually been using any Asus cables, so are there any problems using Asus IDE ribbons cables on other brand motherboards?

They probably just sourced the connectors from somewhere else. As long as the holes line up and it feels snug, I wouldn't worry.

The size of this connector is standard (3M 3417-7000 datasheet IIRC) so I doubt someone can do something too wrong...
And about ASUS... They have bad habits with cables. Like using plugs w/o the rectangular key ledge (floppy drives connecting, yes... *censored* ASUS) and plugs on both cable sides. But an IDE cable is too simple, it's very hard to make it wrongly.