While the terms responsibility and liability have similar meanings, they are used in different contexts.
The term responsibility is more common in general English, and most often used to refer to something that it is your job or duty to deal with. For example, if you are promoted at work then this may involve additional responsibilities. These responsibilities will typically involve being charge of someone or something, and making sure that their work is satisfactory and that what happens to them at work is done correctly.
If you take, accept or claim responsibility for something then you accept the blame for something bad that has happened.
To be liable for something is to be legally responsible for that thing. At work you might be responsible for something that goes wrong. If this should result in legal action, a court might find you or your company liable for the consequences of what has gone wrong. Liability is the legal responsibility for the consequences of your actions or for your failure to act when you should have done something. Whereas responsibility implies some kind of moral duty, liability implies a legal responsibility that might result in you having to pay damages (=money compensation awarded by a court). For example, a patient may be able recover the cost of medical treatment following an accident at work if the employer’s legal liability can be established.
The term liability can also be used to refer to a person’s legal duty to pay taxes, fines or other forms of payment. For example, accountants will try to find ways to help reduce their clients’ tax liability.
In business English, the term liability is also used to refer to the debts of a business, and is usually used in the plural. Liabilities contrast with assets, which are something of value that a person, company or other institution owns. For example, a firm’s accounts might show that it has assets of around $11m and liabilities of around $9m.