If your tree represents a network, diameter tells you the worst-case communication distance. In a company hierarchy, it is how many hops a message might travel. In a file system, it is the deepest nesting.
Diameter also helps you find the center of a tree, which is the node that minimizes maximum distance to all others. That is your best meeting point or server location.
Without knowing diameter, you cannot answer questions like "what is the farthest two nodes can be?" or "where should I place the root to minimize depth?"