Postcards are good for many things-for sharing scenes of a vacation spot, as flashy, quaint or sentimental curios, and to show someone you are thinking of them even though you can't fill up a full letter with things to say.
But postcards have another advantage, at least in the United States: they are cheaper to mail.
(We are talking postcards of normal dimensions. If you want to get all fancy and mail out an extra long postcard, or a circular postcard, you can't expect a bargain on your postage). Since complaining about the price of postage is tied with complaining about the decline in quality of Saturday Night Live as a popular pastime, choosing to limit your epistles to a single paragraph allows you to win a moral victory over big postage. The current prices are 42 and 27 cents for letter and postcard, respectively, and it has been at about this ratio for the past several decades. However, just recently, the United States Postal Service has discontinued dual rates for international postcards. So now, if you are trying to share the beauty of your hometown with your friends in Estonia, you won't be able to save fifteen cents doing it.
Overall, the cost issue is probably not a determining factor in most people mailing postcards. Anyone analyzing the cost structure of the postcard would realize that on a per-word basis, the postcard is actually probably five times as expensive as a normal letter. I don't think that postal cards are actually cheaper for the postal service to handle and ship, either. The difference in weight being fairly negligible, they probably actually end up being more expensive, since it is probably more difficult to design postal equipment that can pick out the address amidst the writing. So the decision by the postal service to charge a lower rate is probably due to the fact that they know that if people weren't getting a bargain, they might be less inclined to pay the full, torturous forty two cents to mail a silly picture with a silly paragraph to Aunt Mabel. Although who knows what the real justification for it is?