Power shall be used for any enhanced capability directed by a person. It includes powers direct, indirect, and material. Direct powers would be talents, skills, and knowledge. Indirect would be ability to command the services of others, be it through respect, persuasion, or threats. Material sources include raw materials, tools, and capital (money).
Responsibility shall be used for some action or function that person is expected to execute. It may be a personal responsibility (eating well, avoiding danger) More often responsibility is used for a function that is a common good for at least a small group of other people.
A police officer has the responsibility to maintain the peace. He has the powers to arrest or harass (fine, etc) those that disturb the peace, as well as training to handle expected situations.
A government has the responsibility to protect it’s citizens safety and provide for the common good. It has the powers to make and enforce laws, maintain armies, and usually tax.
A CEO has the responsibility to keep a corporation functioning and profitable. He has wide powers to command and direct the resources of the corporation towards these ends.
The key concept in all these examples is that power is given to enable to execution of responsibilities. This the right relation. To give power with the hope that it will be use responsibly puts great faith in the character of the recipient. Responsibility without power is helplessness. Power without responsibility is tyranny.