Actuallly you need different variable names for the old and new project id's:
Code:
accept employeeName prompt 'Enter Employee Name to Change Project:'
accept EMP_projectID_old prompt 'Enter Old Project ID for &employeeName:'
accept EMP_projectID_new prompt 'Enter New Project ID:'
accept empID prompt 'Enter the Employee ID:'


UPDATE employee
   SET emp_projectid = '&&EMP_projectID_new'
 WHERE employeeid = '&&empID' 
   AND emp_projectid = '&&EMP_projectID_old';