What is the difference between Sphere of Influence and Hill sphere? Wikipedia's definition of Hill sphere is:


  An astronomical body's Hill sphere is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. 


And then there is the sphere of influence:


  A sphere of influence (SOI) in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is usually used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects (such as moons), despite the presence of the much more massive (but distant) Sun.


To me, a planets Hill sphere and its sphere of influence appear to be the same. But the formulas to calculate the two sphere are different.

So what is exactly the difference between Hill sphere and the sphere of influence?
The two terms are used in answering different questions. 

Hill Sphere: given a large mass (eg Sun) and a small mass (eg Earth), can a tiny mass (eg Moon) find a stable orbit around the small mass? (If the tiny mass goes outside the Hill Sphere of the small mass, no.)

SOI: given two large mass objects and a small object between them, (eg sending a probe from Earth to Mars), which massive object should we use as the origin of the frame of reference? (The small object is within which massive object's SOI?)

As the questions are different, different approximations are used to answer the questions.