I've looked them up in many dictionaries, it seems the two words have the same meaning, right? I don't know when we use each of them properly.
If you are accountable for something, you are in a position that requires you to report to or answer to someone for your actions and decisions.
If you are responsible for something, you may be the one to whom things are reported.
I think with accountability there is someone or some agency overseeing your actions; someone to whom you report.
With responsibility, you are the one in charge. "The buck stops here." One dictionary says when you are responsible, you have the ability to act without guidance or superior authority.
In a nutshell (from Merriam-Webster dictionary):
"responsible" implies holding a specific office, duty, or trust;
"accountable" suggests imminence of retribution for unfilled trust or violated obligation.
So, as you can see:
You TAKE responsibility and you are HELD accountable.
In my opinion, accountability refers to the quality which a person or body has when they are reliable in their behaviour, capable of taking justifiable actions or decisions; it also indicates that a certain (positive) behaviour or attitude are likely to be kept over time, so that you can trust the person or body.
Responsibility on the other hand refers to the fact of being in charge with something, of having control over someone, or having a duty to deal with something. The person who is responsible for such situations may be accountable, but it is not a necessary feature of the person himself.
Here's what I learned as an officer in the Navy - responsibility for a task can be delegated, while accountability cannot. As you may have read in other places on the inter-webs, responsibility may be shared among several people, but there can only be one person who is ultimately accountable ('held accountable') for a given task.