Hydrometer is a device used to measure density of a liquid whereas hygrometer is used for measuring the moisture content in the atmosphere.
Hydrometer of a constant weight sinks more in a less dense liquid. Hydrometer works on the principle of flotation. It sinks more in the less dense liquid so as to displace large amount of less dense liquid whose weight is equal to the weight of the hydrometer. On the other hand, it sinks less in the more dense liquid so that it can displace less amount of the more dense liquid whose weight would be equal to the weight of the hydrometer.