This post adequately explains why one needs a credit card for hotel room reservations, car rentals, and similar. But why do I not need a credit card to rent an apartment? Although I do pay a deposit, that deposit is nowhere near the value of the property (unlike, for example, the deposit I might pay for renting a bicycle). What makes that I can rent some things, but not others, without a credit card?
Because you don't rent apartment for a week. That means, the apartment will likely to be empty (you'll pay much higher rate/deposit for furnished ones), and without any utilities (you'll have to order those in your own name). So the damage you can do is rather limited. Soil the carpets and draw on the walls, mostly. Not something that your deposit cannot cover.
For more significant damage, you'll get sued. Also, keep in mind that renters are usually required to have renters' insurance, to cover the more extreme damages and breakages.
Also, in addition to the deposit, lenders look at your "credit history" - references from previous landlords. Being bad once will make it much harder for you to find a place to live for years later, at least in the US.