what is the difference between a hot water boiler and a steam boiler?
This depand on many things: Hot water boilers are used for heating purpose and others are used for steam turbines in power stations.
that a hot water boiler heat thing up and a steam boiler just add water or sweat them out.
It is not very simple: Hot water boiler takes as input always water ad at output hot water and steam. Steam boiler can take as input water but generally takes a mix of steam and water and is used to heat the mixture until saturated steam and sometimes superheated steam so the output is always as steam.
Edited If a hot water boiler does indeed boil water it too would be a steam boiler. First, it seems that the term boiler is being misapplied to just a water heater. Usually, hot water heaters only heat fairly low-pressure water to below saturation temperature, often to not more than 180F and are not intended to produce steam. They are used for various building and process heating applications. In the other case, the use of the term steam is being used redundantly with boiler. Simple boilers heat pressurized water to its saturation temperature and add more heat to produce the evaporation of the water into steam at a predetermined rate at that distribution pressure. Boilers can be used both for power applications and also for various heating applications. The more complex power boilers have an additional superheat section where the temperature of the steam is increased at the same pressure. This is intended to improve overall efficiency and to avoid condensation and erosion within the power turbine.