What is the difference between horizontal and equatorial coordinate systems apart from the notations (azimuth, altitude) and (right ascension and declination angle)? Please provide some measurements of stars with diagrams if possible and two diagrams comparing the measurements are taken by two different observers in the case of horizontal coordinate systems (since they are supposed to be relative to the observer so the measurements are supposed to be different-at least this is what i am told)
One of the biggest differences between those two is their dependency on observers. The horizontal coordinate system depends on where are you on this Earth, for example, altitude is measured from your horizon it may be different for some other person somewhere else but you will not face this problem in the Equatorial coordinate system, for example, declination is measured from the celestial equator so it will not change whichever location you go!
You don't choose a coordinate system. If you're navigating you take sights to get horizontal coordinates, then look up the equatorial coordinates, and knowing the time, calculate your position. If you're observing some celestial body you start off knowing its right ascension and declination, and a GoTo telescope mount will do the calculations needed to point the telescope in horizontal coordinates, knowing its location and the time.