Stirred or Baked
Custards are liquids that are thickened by the coagulation of eggs (wholes or yolk). They are cooked slowly over low heat until it is thick and creamy, many custards are variations of the crème anglaise with extra steps or ingredients.
Crème Anglaise is cooked on the stove-top (or stirred) they must be cooked in low heat and constantly stirred to stabilize and avoid curdling. You don't want scrambled eggs for your crème. Pastry cream is done like this too, but we finish adding some corn starch to get to the desired thickness.
The baked custards finish the egg coagulation (thickening) in the over and it is done over bain-marie, a water bath. In addition to helping to cook the custard evenly, a bain-marie can also help to prevent the surface of the custard from cracking. Because the water in the bain-marie creates a moist environment, the surface of the custard is less likely to dry out and crack as it bakes.
The Crème Brûlée in the pictures above is a very traditional desert and a baked custard. The top of the custard is caramelized after the custard in the ramekin is done. After baking, cooling and refrigerating the Brûlée is finished with a layer of sugar on the top and caramelized with a blow torch.
Once the caramelization on the top is done the Crème should be readily served for consumption. While the custard can be stored for later consumption, once the caramelization on the top is done it cannot be maintained properly. It could easily get too hard and crack with the cold or loose it's hardness and melt, loosing the characteristic of the desert.