Custard and Vanilla Pudding are nearly the same thing.
However Vanilla Pudding is different from Custard.
Vanilla Pudding or any pudding is a sweetened milk or cream based mixture that is thickened with a gelatinized starch that is usually cornstarch or flour that is then cooked on the stove.
Custards are cream or milk based and is thicker than Vanilla Pudding or other puddings.
So while Custard and Pudding including Vanilla pudding are similar they are different and are not the same thing.
They sometimes taste a bit different as well depending on your tastes.