What advantage does the "blisk" or "bladed disk" type rotor have over the usual rotor system?
Since the blades and rotor disc are a single piece, there are no additional fittings (bolts, screws etc) to attach the blades reducing complexity and weight. The blade root joints are also a source of failure through manufacturing defects and fatigue cracks.
The disadvantage is that any damage to a blade requires the entire assembly to be replaced rather than a single blade.
"Module weight savings of 15% and improved aerodynamic efficiency", according to Rolls Royce.
From a manufacturing point of view, replacing literally hundreds of sub-components (individual blades, etc) with a single part has obvious advantages - though of itself, that wouldn't give the customer any benefit if it didn't also reduce weight and/or improve performance.
The above answers give very good technical reasons for blisks. Less parts means less rotating weight which has a high impact on overall weight.
However, the technical benefits are not the only considerations. As engines age eventually they are worth more in parts than as an engine and the parts are salvaged and sold. The compressor blades can bring in a tidy sum. However some will have erosion and other nicks that will not meet the manual limits for returning to service. So not 100% of the parts removed from a compressor can be salvaged and sold. There are several thousand blades in a modern high pressure compressor. This salvage market is competition to the aero engine manufacturers. The manufacturers would rather you buy a new part from them than a used part from someone else.
Aircraft engines follow the same economic model as razor blades. The initial purchase may be sold at a loss but the replacement parts will be profit drivers.
A salvaged Blisk will have airfoils with erosion and or nicks and dents on the airfoils (this can happen after only a few hours of operation) So the blisk assembly will not be able to be inspected serviceable. The engine operator will need to send the part to the Engine Manufacturer for a proprietary repair. This gives the engine OEM a better control on the used material on the market.
In conclusion, less weight is always good but the blisk also provides an opportunity for the engine OEMs to ensure that they sell parts and make a return on their initial investment.