What is the difference between Sphere of Influence and Hill sphere?
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.