Why does the cart show a different shipping charge from the one at checkout?

The cart page doesn't know the customer's full address yet, so it estimates shipping from Settings → Store Settings → Shipping Setup (your flat or per-product charge, and free shipping above your minimum).

Shipping Zone rates are matched by country, state, city and pincode. They only appear at checkout, once the customer enters their address, under Shipping method. The zone price then replaces the cart estimate in the order summary.

So a difference is expected when you use zones. If you want the cart estimate to be closer, set your Shipping Setup charge to the rate most of your customers pay.