What advantages come with the wing on the Boeing Bird of Prey?
The Boeing Bird of Prey is indeed a unique aircraft, wahich was an internal project rather than an X-Project funded by the government. It was primarily used as a technology demonstrator for low observable technologies.

The Bird of Prey is rather unusual for a wing only aircraft (in the sense that it had no stabilizer) that it was statically stable and required no correcting inputs from a computer. This stability was achieved by the lift generated by the chines in front of the fuselage.

Here is  closer look at the chines.



Source: www.freerepublic.com

The chines produce lift by generating vortices (similar to the delta wing). Basically, the chines produce seperated vortices on either side of the fuselage, which produces lift. According to adg.stanford.edu,


  Rather than reducing the lift of the wing, the leading edge vortices, increase the wing lift in a nonlinear manner. The vortex can be viewed as reducing the upper surface pressures by inducing higher velocities on the upper surface




Image from Effects of Vertical Tail and Inlet/Strake Geometry on Stability 
of a Diamond-Wing Fighter Configuration by Mitchell E. Fossum et al., accessed through http://enu.kz/

To compare how the chine generates lift similar to a delta wing, it is instructive to see the vortex flow patterns on a slender (cropped) delta wing with a chine forebody.



Image from On the Nonlinear Aerodynamic and Stability Characteristics of
a Generic Chine-Forebody Slender-Wing Fighter - Configuration by Gary E. Erickson and Jay M. Brandon, NASA Technical Memorandum 89447, labeling mine.

An example of a delta wing with a chine forebody shows the two vortices being generated from the chines and the highly sweptback wing, similar to the Bird of Prey.



Image from Aerodynamic Analysis of a Generic Fighter with a Chine 
Fuselage/Delta Wing Configuration Using Delayed Detached-Eddy Simulation by Tiger L. Jeans et al, accessed through calpoly.edu 

The forces acting on the longitudinal axis of the aircraft look something like this (I'm not sure of the exact values or locations), leading to a statically stable aircraft.



Profile from bj-o23.deviantart.com, others own work.

The chines also help in reducing the radar cross section, according to dictionary.sensagent.com (attributed to The Radar Game by Rebecca Grant):


  A standard leading edge extension joins to the fuselage at an angle while a chine is an extension of the curvature of the fuselage. Therefore chines avoid presenting corner reflectors or vertical sides to radars.


This design has since become standard in (Boeing) UAVs like X-45 etc.

Another feature is that the fuselage sides are parallel to each other, helping in reducing the Radar Cross section by reflecting the radar waves away from source.



Source: gizmodo.com

The wing design is rather unusual, a polyhedral section with dihedral (upwardly angled) inboard sections and anhedral (downwardly angled) outboard sections, which helped in stealth and stability.



Source: soldiersystems.net

The wing has a rather high sweep angle (~65 degrees) for a low speed aircraft, in order to improve stealth.