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;
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
The Cambridge Dictionary defines big as "large" and large as "big." There is no difference in the implied size, which is relative to other quantities; large is simply larger than medium, and much larger than small. It is relative to the norm or expectation of the size of that item, or the size of the item being considered by the speaker/writer. Idiom often determines the usage. We don't say small, medium, and big; we say small, medium, and large. Large is rooted in Latin (though it does not have a strong Romance flavor); big (etymology uncertain) appears to be Germanic (Old English, Scandinavian). If you write a poem with the two words and speak it aloud, you can sense the difference in flavor. Though both are only one syllable, they have a different quality. The word big expresses some emotion (such as surprise), attitude, or reaction toward the large size: Big deal! Big girls don't cry. Bright lights, big city ("Look at all them big buildings!"); whereas large seems more objective. From the Cambridge Dictionary: She had a big pay rise. (I contend that this expresses a reaction to the large size.) We need a larger house. One could say We need a bigger house too, but the former sounds more formal, with no feeling added. We have a big family now, and we need a bigger house and more bathrooms! Big can more often refer to nonmaterial elements: I have big doubts. Large is more quantifiable in material terms: I come from a large family of ten siblings. In formal writing (such as a letter used to apply for a job) one would use large (unless big is the idiomatic choice); I worked for a large company, not a big company. There's not a great difference between these words in terms of dictionary definition, but most native speakers instinctively observe the nuanced distinctions.