The potential difference across a component is the energy transferred for each coulomb of charge that passes through it. The current is the number of coulombs passing each second. So the energy transferred each second, which is the power, depends on both. For any circuit device, power = current × potential difference, or P = I × V.
A bigger potential difference means each coulomb transfers more energy. A bigger current means more coulombs pass each second. Either change gives more power. Doubling the potential difference with the same current doubles the power, and halving the current with the same potential difference halves the power.
Worked example: a 12 V lamp with a current of 2 A has P = 2 × 12 = 24 W. A kettle on 230 V mains with a current of 10 A has P = 10 × 230 = 2300 W. If the current were 5 A at the same potential difference, the power would be 1150 W. On the same 230 V mains, a device with a higher power rating needs a larger current.