Deflection of a beam

It is because we want to ensure the servicibility of the beam. Do you want to stand in a room with a visible deflection, even though you are told that the room is 100% structurally safe. Also, large deflection would induced cracks in the structural element. Large cracks would cause water leakage. British standard defined the max. deflection limit to ensure the servicibility of different structural elements.

totally agree with isistsetse's answer

1 more thing to supplement is, excessive cracks will decrease the durability of reinforced concrete because CO2 & other acidic gases will decrease alkalinity of concrete, hence increase corrosion rate of reinforcing steel.

Of course, there is a relation between deflection and moment capacity, and crack widith. However, they are not the purpose to determine the deflection. You can calculate the moment capacity and crack width directly by other method.

For building design (not water retaining structure design), deflection means serviceability of the structure. It will affect the service of the structure, but not the structural safety.

let me simplify a little for you,
To determine the deflection can tell you how much the beam can bend. which will tell the engineer to modify their design in order to make the design safe to use.