How can bromine water be used to distinguish between benzoic acid and salicylic acid?
Salicylic acid is partially activated towards electrophilic aromatic substitution by the hydroxyl group, and slightly deactivated by the carboxylic acid. Overall, it is activated towards the substitution. Benzoic acid is deactivated by the carboxylic acid, so overall, it is deactivated towards substitution. Hence, salicylic acid will react without the need of a catalyst (and upon decarboxylation, the reaction will actually become faster), whereas benzoic acid needs a catalyst, such as $\ce{FeBr3}$, to react. 

Hence, salicylic acid will react with bromine water, but benzoic acid won't. Physically, bromine water is orange due to $\ce{Br2}$, but if the bromine reacts, then the orange color will no longer be present. Thus, the solution will turn colorless upon contact with salicylic acid, and will stay orange in contact with benzoic acid. Furthermore, depending on the scale and concentrations, you may also observe the evolution of $\ce{CO2}$.