The terms box spring and foundation are used interchangeably and both refer to the base that you put into a bed frame to support the mattress. But the two terms used properly can refer to 2 totally different things.
Box spring originally referred to a group of springs encased in a box and covered with fabric. They were first built from metal, but are now mainly constructed with wood, and both types exist on the market today. Many box springs are specifically constructed to correspond to a given mattress, and the two are sold together as a set. Box springs sold in a set do not offer the proper support to other mattresses. Box springs are meant to give, or compress, when weight is applied. So when you lie down on a mattress set, the springs in the mattress compress and, where the weight is greater, the box spring also compresses.
A mattress foundation looks just like a box spring on the outside, a box encased with fabric, but it is constructed to offer a firm, flat surface to the mattress that will not give or compress. Foundations are the requisite base for a latex or memory foam mattress because they offer a surface that will not compress under body weight.
The softer top layers of a latex or memory foam mattress are meant to conform to and absorb the entire weight of your body while the underlying base foam layers offer a firmer support. When a foam bed is used with a box spring, the base foam layer cannot do its job. Rather than pushing back against your body to provide support, the base foam layer pushes the box spring lower. You end up sleeping in a concave pit with your hips too deep in the bed and your feet and head too high.