The example from the article is people trying to figure out when the group should all go to dinner; no one actually cares the end result, the only that is needed is agreement. This isn't free in many real-world problems. Knowing that you have correctly stored the final value is sometimes required. Vector clocks only lets you fix the problem after it has occurred.
Dave is not available to resolve the conflict at time 4
Dave sorted out a conflict at time 3 in all versions of the the story. The problem is that nobody else knew which message was from time 2, and which was from time 3. Vector clocks fix that problem.
Now in this particular case you could use a normal clock, but in more complicated scenarios that can lead to mistakes where vector clocks still work.