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Standby:- All components are in power saving mode, but still receiving power. So when the computer see any activity (like mouse moving, pressing key of keyboard, it instantaneous start. It do drain battery/main power but is less than the what is used for normal working. It is good only if you leave your system for few minutes.

Hibernate:- When system hibernate, it copies all the memory data on hard disk (i.e.if all application open are saved as it was when you hibernate). But after saving all memory data, it can completely shut off (i.e. no power required, no battery/main power drain). But you have to start your computer again and it will start instantaneously by copying saved memory data to your system's memory (without going through steps of booting operating system or starting those applications). It is good if you want to leave system for few hours or days.