The restored armour is the only effect you can see, but behind the scenes it also switches armour type, so it takes less damage. First veterancy level takes 90% damage, second level takes 80% damage (subject to change of course). This is currently universal across all vehicles, but can be changed by adding more armour.ini entries.
Oh, and also some vehicles will get upgraded weapons as they get promoted. You might have noticed in the video that the apoc took down the last pillbox VERY quickly, this is because its weapons had been upgraded when it got promoted the second time, and did a lot more damage per shot.