What is the difference between an academic department and program in US colleges/universities?
I am not sure everyone uses the words the same way, but here is the difference at schools I have been at:


A department is a collection of faculty who are organized into a unit for management and course planning purposes. Some departments have faculty from just one area, while others have faculty from many areas. The division into departments is used in particular for hiring and tenure decisions, which often begin with at the department level. Several departments are collected together to form a division, school, or college, which is the next level up the management hierarchy. Different schools use different divisions, but the smallest one is almost always called a department. 
A program is a collection of courses that lead to a particular degree or certificate. These courses (ignoring general education) may be taught by one department, or by many. For example, a degree program might be "bachelor's of science in mathematics" or "bachelor's of arts in management".  The mathematics program will require predominately courses from the mathematics department, while the management program may require courses from the management department, the accounting department, the economics department, and maybe a computer information systems department also. 


Some departments offer a single degree program, while other department offer multiple degree programs. On the other hand, some degree programs are interdisciplinary and are offered jointly by several departments in collaboration. 

Each faculty member will work for one or more departments; these assignments are called "appointments". A student will not be part of a department in the way that faculty are, but a student may "declare" one or more degree programs, and in that way each student is typically "advised" by faculty of one or more departments.