What's the difference between "big" and "large"?

What's the proper way to say: a large family or a big family? What's the difference between them?

Nothing really. In English you tend to get a lot of words that mean the same thing, sometimes there are historical or poetic reasons for choosing one word — but not in this case.

Other than big being a much more common word and large sounding more refined there aren't many areas where you would use one over the other for purely grammatical reasons.
Note that big can also mean "major or important" — so big decision, big spender.

Big and large overlap for sizes and numbers;

However

big can mean 'important'

• a big family ~ an important family / a family with many members

• a large family ~ a family with many members

• a big event, an important event / an event with many people involved

• a large event - an event with many people involved

big can mean 'grown up, old enough, mature'

• a big boy - a grown up boy, old enough / a big-sized boy
• a large boy - a big-sized boy