Helicopters use a cyclic control to tilt each blade differently through the rotor's revolution. This is quite a complex mechanism. What is the advantage of this vs tilting the whole rotor plane? This seems to be a much more simple solution.
Tilting the rotor doesn't do much, in fact the body will be tilting more than the rotor. The gyroscopic effect will work against you.
The cyclic increases lift on one side of the rotor which will tilt the rotor. Which will then move the craft in the correct direction.
In other words the cyclic system gives a more direct response to input.
Rachet Freaks answer is good, but I feel a bit incomplete.
You ask why it needs a cyclic control. Well, you need a control to tilt the disc, either directly or by flying it into position, so let's call that control the cyclic control (since it affects the rotor cyclically). So whatever your mechanism, you need a cyclic control to tilt the disc.
So what do we mean by tilt the disc? It's splitting hairs a little bit, although precise, to distinguish between tilting the disc, and causing a cyclic change in lift to each blade such that the rotor flies into a tilt. The reasons for flying it into a tilt are as Rachet Freak states.
You then state that it's complex, so let's examine physically tilting the disc versus cyclically altering the pitch of each blade.
Cyclically altering blade pitch: The power shaft from the rotor gearbox passes through the middle of the swashplate and is attached to the rotor. The bottom of the rotating rotor is the rotating part of the swashplate which acts as a bearing surface rotating on the lower part of the swashplate which is non-rotating and fixed to the fuselage. The blades are attached to the rotating swashplate with pitch links which can move to turn the blades in their cuffs. Now all we need to do is to attach the cyclic control to the non-rotating swashplate so that we can tilt it. In turn, this will tilt the rotating swashplate and the pitch links will transmit this to the blades, causing them to rotate in their cuffs, altering the angle of attack and therefore lift.
(By KoenB (blender drawing) [Public domain], via Wikimedia Commons)
Physically tilting the disk: The first half of this is the same. Just remove the pitch links so that the disc will physically follow the swashplate. All good so far, and the only complexity you have removed are the pitch links attached to the blades to rotate them. But now you have a problem. You can only increase or decrease total lift by varying the power applied to the rotor from the engine and therefore the total rotor thrust. The easiest way to solve this is to put the pitch links back and allow the blades to rotate in their cuffs.
Therefore, trying to physically tilt the disc needs the same complexity as cyclically varying the pitch and loses a lot of the utility of the helicopter.
The earliest autogyros by Juan de la Cierva did this, tilt the whole rotor. But when the autogyros became bigger, the control forces became too heavy and that is when he designed the feathering mechanism, so that aerodynamics could take care of it by using blade flapping.
It's actually quite beneficial to not have to tilt the whole rotor: the rotor axis is beefy and it has gearboxes connected to it. Yes the feathering mechanism is complicated, but a construction to allow the rotor axis of let's say a CH-53 to tilt, with its three engines and gearboxes connected to it, has its own complicated problems to solve.