A GC is a license to be in the US, so when I am in the US I carry it with me. What's the difference?
Three important differences:
1. Everybody who drives is supposed to carry a license, whereas some 90% of the population (citizens) aren't required to carry any immigration documents. That makes it useless for detecting illegal immigrants on an ad-hoc basis, because if a person is randomly stopped the lack of a green card in the person's immediate possession doesn't give a clue as to whether they are illegal or not ... because 90% of the people you stop aren't going to have documents and aren't required to have them. So you're ultimately going to have to revert to other means like guessing based on how they look and speak, or taking their word for it, or looking up their information on a computer based on some other ID they might have on them.
If the driver's license rule had a similar condition where only a small subset of drivers have to carry a license (for example, only people who were born outside a given state have to carry a license when driving within the state) the requirement to carry the license on one's person would be just as useless for detecting unauthorized drivers. On the other hand, the requirement to carry immigration documents actually is useful in places where everybody is required to show them, like when arriving from overseas travel or applying for a Social Security card.
2. Driver's licenses are much quicker and cheaper to replace than green cards.
3. The requirement to carry around the green card is much more onerous and causes a much greater risk of loss or theft than carrying a driver's license, because the obligation to carry a license only applies to when one is actually driving; not when you're walking down the street or riding a train or being a passenger in a car or swimming at the beach. Whereas the green card rule applies to everywhere you go in the US and everything you do.