1.What's the difference between different and differential?And how to use them correctly?
2.What's "gut-driven" mean?
Different is a very common English word that every English speaker will use thousands of times in his life.
Differential means of, or relating to a difference. It is a word that most English speakers will never use except when refering to mathematics (e.g. differential calculus) or mechanics (e.g. the differential gear in your car's rear axle).
The word gut in many instances (your example is one) means intuition or emotion rather than logic or observation.
A "gut-driven" decision is one that is made with your feelings rather than your intellect.