'Big' can mean 'important', for example: 'Buying a house is a very big decision'.
It can also be used in informal situations to mean 'older', for example: 'He's my big brother'... as well as 'successful' or 'powerful', for example: 'York is a big tourist destination'.
Also in informal situations, we can use 'big' to mean 'doing something to a large degree', for example: 'She earns a lot of money, but she's also a big spender' - OR... 'I'm a big fan of yours'.
'Big' is used in a lot of fixed phrases, and because these phrases are fixed, to change 'big 'to 'large' would sound wrong. Examples of fixed phrases using 'big' include: 'It's no big deal' - it's not really important. 'I have big ideas for this house' - impressive plans for the future. 'She's a big mouth' - a person who can't be trusted to keep a secret. 'He's too big for his boots' - too proud of himself.
There are also some fixed phrases using 'large'. Examples include: 'The prisoners are at large' - they have escaped and may cause harm. 'She's larger than life' - more exciting or amusing than most people.
Finally, quantity words.... 'large', more often than 'big', is used with the following quantity words: 'a large amount', 'on a large scale', 'a large number of', 'a large quantity of', 'a large proportion', 'to a large extent', 'a large percentage of', 'a large part of', 'a large volume' and 'a large area'.