If you take responsibility for your own actions, you show accountability. Stepping up and admitting it when you break something shows accountability.
Accountability is a noun that describes accepting responsibility, and it can be personal or very public. A government has accountability for decisions and laws affecting its citizens; an individual has accountability for acts and behaviors. Sometimes, though, taking accountability means admitting you made a mistake. Punishment may result, but accountability shows ownership and a willingness to admit mistakes.