‘Freedom’ is both a quality and a condition. ‘Liberty’ is only a condition. When you can act as you wish without any restraint, you are said to enjoy freedom.
Freedom is a state where you are free. liberty is absence of any restraint.

When someone is set at liberty, he is freed from restrictions.

In several contexts, 'freedom' and 'liberty' are interchangeable. Freedom of speech or liberty of speech, freedom of the press or liberty of the press and so on. Freedom has more positive connotations than liberty.

The Prime Minister was given the freedom of the city.

'Liberty' is not used in this context: You speak of the 'freedom of the countryside', not the 'liberty of the countryside'.

‘I was given the freedom of his house’ means I was allowed to use his house as if it were my own. Here again you can’t use ‘liberty’. ‘To take liberties’ means to be a little too free or familiar in action or speech.

He gives freedom to others but you can't take liberties with him.

You can talk about ‘freedom from want’ but not ‘liberty from want’. Also we talk about ‘freedom fighters’ and not ‘liberty fighters’.”

“What is the meaning of ‘throw out the baby with the bath water’, sir?”

"When you give a bath to the baby, you finally throw out the dirty bath water. It would be awful if you threw out the baby also along with the water. When you reject something very important along with the unimportant, you are said to throw out the baby with the bath water.
Industrial pollution is bad. But if you say that industries should be eliminated to get rid of industrial pollution, it would be like throwing the baby out along with the bath water.