Different cars behave differently to some extent and you may need to use more throttle (more gas) than you did in your instructors car and it may need to use a lower gear to climb hills. It takes time to get used to a different car, especially for the first time. I learned to drive almost entirely in a car that was 15 years old at the time( and from the 1950's
) I had a few professional lessons in a brand new driving school car that was so smooth,powerful and easy to drive in comparison it was a doddle . I'm proud of the fact i actually passed my test in the old banger
But I'm intrigued that you expect to drive the car without using the handbrake much. Its bad practice not to apply the handbrake,and go into neutral) every time you stop ,even on level ground, even in stop start traffic. Some drivers might not ,but its a bad habit. And hill starts most definitely need you to use the handbrake, apply significantly more gas than normal and slip the clutch to some extent to smoothly take up drive. And be in the correct gear for climbing any given hill. Your instructor may have avoided such hills as they can be a challenge for novice drivers. With practice it becomes easier, and shouldnt put much undue strain on the clutch, if you do it properly and the car is running as well as it should. Its unavoidable that a car used frequently in hilly districts or heavy traffic(or a driving school car driven by novices) might wear its clutch out sooner than a car that does mostly motorway miles.
But you should check your car is indeed running properly and 'pulling' as well as a 1.4 litre jazz should. You could ask a mechanic to test drive the car to see if the car is performing as expected, or any experienced driver who is insured to drive the car could probably tell if somethings not right.