Energy transfer - the charge carriers take energy around the circuit. When they go through an electrical component energy is transferred to make the component work.
No appliance transfers all energy completely usefully. The higher the current, the more energy is transferred to the thermal energy stores of the components (and then the surroundings).
A higher power doesn't necessarily mean that it transfers more energy usefully. An appliance may be more powerful than another, but less efficient, meaning that it might still only transfer the same amount of energy (or even less) to useful stores.
A battery with a bigger potential difference will supply more energy to the circuit for every coulomb of charge which flows round it, because the charge is raised up "higher" at the start.