How is the organization of school systems in the US different from state to state?

Under the Tenth Amendment, states have priority in setting educational policy, and most education policy is set by state governments.

School systems in each state vary in a number of different ways:

Some states have "public school choice", in which students can choose to attend a different school than the one in their neighborhood; in some states such as Arizona students can even shift districts if they want, while in others such as California they must stay within their own district. Other states such as Alabama do not allow public school choice at all and require students to attend the school closest to where they live.

Most US states allow charter schools, which are funded by the government but operated independently under more flexible rules. A few, however, do not, such as Maine.

A few states allow students to use vouchers to pay for private school if they do not attend public school, but most do not. Often states that don't allow charter schools do allow vouchers (again Maine is an example).

The mandatory starting age varies between states from a low of 5 to a high of 8, with most states setting it at 6 or 7. States also vary in when students are allowed to drop out, as early as 16 or as late as 18. Some states require kindergarten, but most do not.

States also vary in their curriculum standards, their homeschooling regulations, and a number of different ways. I've linked an index of public policy research that has spreadsheets providing comprehensive state-by-state comparisons.