I think it is correct to assume that the hotspot tool would be a bit same as the density tool when no analysis field is specified and the point counts themseleves are taken as values to check clusturing. What makes it different from density tool is that it checks clustering of specific values of points for example if points represent cities and you input a schools field in the hotspot analsyis field then it will check if the cities with more schools or less schools cluster or not but it wont check if the cities themselves cluster which the density tool would do. So basically it checks clusturing of specific values (schools in this case) inside the points (cities in this case) as opposed to just number of points.