In order for the billing and shipping info to copy over correctly they rely on having the default names for the form elements. If you have changed either the billing or shipping name of this element it can cause problems. Here are the names that are used internally in the function:
var billingFields = new Array('firstname', 'lastname', 'email', 'phone', 'fax', 'street1', 'street2', 'city', 'state_province', 'postcode', 'country');
var shippingFields = new Array('shipping_firstname', 'shipping_lastname', 'shipping_email', 'shipping_phone', 'shipping_fax', 'shipping_street1', 'shipping_street2', 'shipping_city', 'shipping_state_province', 'shipping_postcode', 'shipping_country');
If you have any further trouble with this please post back with a copy of the form on the checkout page so we can take a look.