Take the case of a box sitting on a table. In an introductory physics course, we
Sure there's a real difference. The first account implies that the box is being compressed vertically because gravity acts on all its parts (molecules) whereas the opposing force is acting on its bottom surface (where it touches the table). The second account implies that there is no such compression, that the box, even if it is somewhat elastic, has the same height when it is sitting on the table as when it is floating in outer space. The first account -- correctly -- implies the opposite: that the (not perfectly rigid) box is slightly less tall when it is sitting on the table than when it is floating in space.
The first account is also more elegant in this sense. Suppose the table is forcefully kicked out from under the box so that the box starts falling. The first account can easily explain this by pointing out that, with the table out of the way, the gravitational force now acts unopposed. The second account has to say -- oddly -- that the kicking away of the table somehow brings a gravitational force into existence.