The amount of blade area determines the total lift available from the rotor. A low-solidity rotor stalls earlier than a high solidity rotor, all other things being equal (the factor is called Aerodynamic blade loading. It is coefficient of thrust divided by solidity). Thus, more blades means more total lift available (more G's or more speed at 1 G).
The disk area determines the downwash, because the disk area is the size of the "pipe" through which the rotor accelerates the mass of air - the bigger the "pipe" the more mass of air acted upon, and thus the less velocity needed to be imparted on the air (momentum is mass times velocity - the less mass of air, the more the speed you need to induce to get an amount of lift). For this reason, bigger rotors make less downwash than smaller rotors - when they are both lifting the same load.
Tied to this is the fact that Power is used to turn the rotor, and the bigger the rotor, the less velocity it needs to induce on the air, so the very much less the power it needs to make the lift. Bigger rotors need less power to lift a load.
So, to lift a big load with a small engine, get a bigger rotor. To maneuver a helicopter, get more blade area.