There is no principal difference between energy in classical and quantum physics. It is just a useful concept which helps physicists to calculate outcomes of experiments. Practically, however, it looks quite different in the 2 cases.
Classically, the energy of a system is defined as the amount of work done by forces to bring it to its current state. Such definition is not very useful in quantum mechanics, because notions like force or trajectory don't make much sense there. Actually, energy is a concept which proved to be extremely useful in quantum mechanics. But one still has to be careful to make the quantum definition agree with the classical one when the quantum effects are negligible. That is, if you push a trolley on 1 meter applying a force 1 Newton, then even if you calculate the outcome using time-dependent perturbation theory of the Schroedinger equation, you must get that the energy of the trolley increases by 1 Joule.
In QM, the energy is essential mostly because the energy operator H (hamiltonian) governs the time-evolution of any system through the Schroedinger equation. Energy can then be defined as the expectation value of H.
