How does the family budget differ from the state budget?

Family and state budgets have different scales. Not only in terms of the total amount, but also in terms of the number of various types of income and expenses.
The family budget usually takes into account the flow of money between the family and the world. Parents are paid outside the family, purchases are made outside the family. There may be pocket money within the family for children, but the share of such expenses is significantly less than external ones.
The state budget is usually associated with the flow of money within the country. Taxes are collected within the country, the salaries of those who receive it from the budget are paid within the country, roads are built inside the country. Yes, there are also foreign trade operations, but for most countries they represent only a small part of the budget.