After you selecting a date from the date-picker the input box does not get the focus, so when clicking on the other input box there is no focusout event on the first input...
Try this:
$(function () {
$("#newDepBirthDate").datepicker();
$("#newDepBecameDep").datepicker();
$("#newDepBecameDep").focusin(function () {
$("#newDepBecameDep").val($("#newDepBirthDate").val());
});
});