Very good question.
Responsible can mean in charge of getting something done. For instance, you may ask, "Who is responsible for dealing with customer complaints?" and you wouldn't use accountable there.
After something bad has happened, someone can 'take responsibility for it', i.e. admit that it was their responsibility to deal with it and they failed. This may be the person who was actually responsible, or their boss, or boss's boss, who set up the conditions that allowed the mistake to be made. Again, you wouldn't use 'accountable' in those sentences.
Accountable is often used for problems. You can be responsible for good things, and take pride in how much responsibility you have. Accountable is usually used in the context of mistakes, errors, and losses. Also, it is usually used when no specific person is mentioned. For instance, "The problem with this company is that no one is accountable for budget overruns."