Aren't both methods used to increase angle-of-attack (AoA)? Blade flapping will change the blades level, while feathering will change the pitch, but if both increase AoA why not just use one method?
The collective pitch control, or collective lever, is normally located on the left side of the pilot's seat with an adjustable friction control to prevent inadvertent movement. The collective changes the pitch angle of all the main rotor blades collectively (i.e., all at the same time) and independent of their position.
That's what you referred to as feathering, notice it says "independent of their position". It's for collectively increasing or decreasing of the lift on all blades—the engine will keep the RPM constant.
Using the collective in level flight would cause a climb or descent, while with the helicopter pitched forward an increase in total lift would produce an acceleration together with a given amount of ascent.
Flapping and feathering are both rotations about a hinge, with each movement having its own axis of rotation, as shown below in a figure from Raymond Prouty, Helicopter Performance, Stability, and Control. It is a top view of a 2-bladed rotor with hinge offset.
The feathering axis allows the blade to be rotated lengthwise. The pilot initiates this movement, via the cyclic and collective control stick, and the Swash Plate transfers this input so that the blades rotate about the Feathering Axis.
The Flapping Hinge allows the blade to rotate up/down. They could point straight up - in normal operation the rotation of the blades produces centrifugal forces that are higher than the lift forces, and the blades point more or less horizontal. The Flapping Hinges are there for counteracting lift dissymetry.
Flapping was discovered as a cure for rolling over when airspeed increased: a blade moving forward has more lift than a blade moving aft. The flapping hinge allows the forward moving blade to move up, effectively reducing the Angle of Attack. Likewise, the aft moving blade descends, increasing the AoA. The effect was discovered by Juan de la Cierva when he constructed his autogyros in the early 20s.
Flapping, the vertical up/down movement of the blades, is not directly controlled. What a pilot controls is the blade feathering or pitch angle. Increasing feathering / pitch generally increases the aerodynamic forces on the blades, which changes the flapping. Unless the rotor and blade are infinitely rigid, which would cause other problems, flapping will be a (often useful) side effect of feathering.
All else equal, higher feathering / pitch at a certain azimuth leads to higher flapping 90 degrees later in the azimuth. This link explains the 90deg offset in technical details: Helicopter Flap Dynamics