They have a higher energy density than most other types of rechargeables. This means that for their size or weight they can store more energy than other rechargeable batteries. They also operate at higher voltages than other rechargeables, typically about 3.7 volts for lithium-ion vs. 1.2 volts for NiMH or NiCd. This means a single cell can often be used rather than multiple NiMH or NiCd cells.
Lithium-ion batteries also have a lower self discharge rate than other types of rechargeable batteries. This means that once they are charged they will retain their charge for a longer time than other types of rechargeable batteries. NiMH and NiCd batteries can lose anywhere from 1-5% of their charge per day, (depending on the storage temperature) even if they are not installed in a device. Lithium-ion batteries will retain most of their charge even after months of storage.
So in summary, lithium-ion batteries can be smaller or lighter, have a higher voltage and hold a charge much longer than other types of batteries.