Why do high efficiency water heaters require outside air vents and drains?
Low efficiency water heaters are low efficiency because their exhaust gases are very hot. Since the gases are hot, they will rise up a chimney flue with ease, but this also means you lose all that energy and warm air up your flue. Older heaters tend to be in older homes where there are enough drafts and air leakage to not worry too much about ventilation. One other thing: the hot air also carries also the moisture produced by combustion up and out the chimney.

High efficiency heaters try to reduce the temperature of the exhaust gases as much as possible in order to recover as much energy as possible. But as the exhaust gas is almost room temperature, it won't rise and thus has to be forcibly expelled. The moisture that is produced by combustion will also condense in the cooler air and will have to be drained out of a separate drain line.

High efficiency furnaces are also designed to be used in high efficiency houses that are well sealed, so you need to bring in fresh air lest you run the heater too lean and produce carbon monoxide. Another reason to bring in fresh outside (cold) air is that it is a waste to use the inside (warm) air for combustion and blow it outside the house.

Heat exchangers are used on the intake and outtake gases which further increase efficiency. This necessitates both intake and exhaust vents.

Thanks to commentator keshlam for additional material that was brought into the answer