The main difference between formally smooth/smooth and formally étale/étale is that the formal versions don't need to be open maps. Chevalley's theorem on constructible topology says that a locally finite presentation map is an open map for the constructible topology, and if in addition you include flatness, you can show that it is an open map for the specialization topology. A map is open for the Zariski topology if and only if it is constructibly open and specialization-open.
As a consequence, smooth/étale maps (being flat and locally of finite presentation) are (universally) open maps.
There are other reasons involving Noetherian approximation to include finite presentation as well. Formally smooth/étale maps don't descend along inverse limits, while smooth and étale maps do. (In the qcqs case, that is).