Apply the rotation
Apply the rotational difference between to vectors to the pose bone, rather than setting the pose bones rotation with it.
The rot diff has an axis and angle. Applying the rotation rotates the pose bone, which in this case is angle 54.7 degrees about axis (-1, 1, 0).
import bpy
from mathutils import Vector
up = Vector((0, 0, 1))
dest = Vector((1, 1, 1))
R = up.rotation_difference(dest).to_matrix().to_4x4()
pb = bpy.context.active_pose_bone
pb.matrix = R * pb.matrix
pb.matrix = R @ pb.matrix # 2.8
To set the bone absolutely from the vector, can use Vector.track_to_quat(to, up) For bones track to the 'Y' axis, along the bone. I have arbitrarily chosen 'Z" as up.
>>> v = Vector((1, 1, 1))
>>> q = v.to_track_quat('Y', 'Z')
>>> C.active_pose_bone.rotation_quaternion = q
>>> x = C.active_pose_bone.tail.x
>>> x * x
0.33333332136784577