I am currently trying to study aircraft performance and I have some difficulties distinguishing between rate of climb and climb gradient.
Can someone explain what are the differences between the two of them?
A climb gradient is a geometry problem -- the relationship of two points in 3-dimensional space... to get from one to the other you gain X' in Y NM, so you have X/Y feet per nautical mile as a climb gradient. Or it can be expressed in percent slope -- the two terms are different ways of describing the same thing.
A climb rate is simply speed in the vertical dimension, X feet gained per minute. It is typically read on the Vertical Speed Indicator.
The two come together when the discussion turns to "what climb performance is required to clear an obstacle." The surveyors determine the geometry between a starting point (such as, the end of the runway at 35' AGL) and an end point (an obstacle at some distance from the end of the runway with a given height, plus a prescribed amount of clearance). Then they publish this as a required climb gradient which will ensure that you'll cross the obstacle with the required clearance. They don't know, or care, what speed you'll be flying; they've surveyed the geometry of the runway and obstacle and applied the regulatory 'required clearance', so their work here is done.
For the pilot in the cockpit, though, I don't have a way to determine my climb gradient at a glance. Performance computers will do that and typically work the calculations to determine how much weight I can take off with, in order to have adequate climb capability, and at some lighter weight they may tell me what my climb gradient will actually be. But it's not a value that I can read off of an instrument as I'm flying.
For a given groundspeed, flying exactly "this" climb gradient will yield "that" vertical speed... define any two and the third can be calculated. So you will often see charts that tell you at various groundspeed values, what vertical speed you will need to maintain in order to achieve the necessary gradient. So the two concepts are related, but nevertheless distinct.
As the not-really-a-duplicate question shows, the airspeed that will give the best climb gradient is not necessarily the airspeed that will give the best climb rate. Normally those are referred to as Vx and Vy -- best angle of climb speed (best gradient -- most vertical distance climbed per distance of forward travel) is Vx, and best rate of climb speed is Vy (best rate of climb speed -- most altitude gained per unit of time). The Vy speed is typically faster than Vx.