The use of Quaternion rotation is to avoid the gimbal lock problem with the Euler method. Usually if your object only needs to rotate around one axis or two Euler would do the job. Like a wheel or a wobbling top. But when you have to animate a fighter jet that rotate all around your scene you may encounter the gimbal lock. To understand gimbal lock there is a great youtube video here.
There are also gotchas in quaternions, it tends to solve towards the shorter path. Meaning if you were trying to rotate say 270 degree clockwise, quaternions would solve by rotating 90 degree counter clockwise. So to solve this correctly you need to add a in between keyframe to use quaternions properly. As described in this video here.
WXYZ are interdependent. Therefore we should never modify them in their values entry box individually. To modify the object rotation simply use R and specify the axis you would like to rotate the object in. With that Blender will translate the rotation on screen into quaternions value if quaternions is the selected solving method.
ARMATURE & BONES
When you are working with armature or bones. In edit mode you will firstly be positioning those bones with Euler. When you switch to pose mode and actually animating, Quaternions are used for solving the transformation of your character's limbs. Often in a rig we use IK bones (Inverse Kinematics), and the IK will drive the armature and in turn quarternion will solve the positions of the bone. We will have no need to edit any of the WXYZ values manually.
Quaternions values are not in degree. Euler uses degree as it's unit. The math behind quaternions are much more complex the 4th dimension (W) allow quaternions method to avoid the gimbal lock of the 3 dimensions that Euler is limited to.