How are the two approaches of understanding gravity, the so-called 'Emergent/entropic gravity' and 'Quantum gravity' (String theory, LQG etc) differ from each other? I am very curious and will appreciate if somebody can sketch a summary.
Quantum Gravity is the description of gravity according to the laws of quantum mechanics. It uses fundamental principles of quantum physics (like quantization for example) and applies it to gravity. Quantum gravity is needed in situations when you cannot just ignore quantum effects (even though you are talking about macroscopic objects). An example would be the quantum effects found near black holes and in the early moments of the universe, jus after the Big Bang. Quantum gravity is still an unsolved mysteries, but candidates include string theory and Loop Quantum Gravity, which tries to quantize spacetime itself. But both are yet to be experimentally verified.
Emergent / entropic gravity is a rather recent theory. Emergence means that you observe some phenomena in an object at a macroscopic scale, but don't observe it when you look at tiny parts of the object. According to this theory, gravity is an entropic force, i.e. it arises as any system's tendency to increase its entropy.
The difference between quantum gravity and emergent gravity is that quantum gravity tries to describe gravity as another fundamental force, which can be explained by the fundamental principles of quantum mechanics and quantum field theory. It also predicts that there should be another fundamental particle, the graviton which is the particle 'communicating' the force of gravity. But entropic gravity does away with the idea of gravity as a force entirely. According to entropic gravity, gravity is just an emergent phenomenon (it is observed for large, macroscopic objects, but not for microscopic constituents of the large object). This theory actually comes from a consideration of the Holographic principle.
So the difference in philosophies between the two theories is: Quantum Gravity tries to describe gravity as a fundamental force (like electromagnetic, strong and weak forces), while Emergent/entropic gravity does not consider gravity a force but rather an phenomena observed in macroscopic objects due to the statistical behaviour at microscopic scales.