In which contexts might students want to determine the chloride concentration of a solution?

The salinity of sea water is one obvious context but this can be extended to find the extent to which salt is found in rivers. Food and drink also provide interesting contexts and opportuntities for student investigations into the chloride ion content, for example, of cheese, butter or bottled water.

Are there other methods that could be used to find the concentration of chloride ions in solution?

Yes, there are several methods that can be used. Volhard's method involves adding excess silver nitrate solution and then a back titration with potassium thiocyanate solution using an iron(III) salt indicator. Fajans' method, on the other hand, uses an adsorption indicator such as fluorescein. Alternatively, the chloride concentration can be determined using a gravimetric method in which the precipitate of silver chloride is weighed or using instrumental methods such as conductivity measurements or an ion selective electrode. These methods are referred to in the Analytical chemistry course, to which you can subscribe.

Are there any drawbacks to using the Mohr titration?

The main drawback to using any titration with silver nitrate is that it is expensive. For this reason it's best to use fairly dilute solutions.