Unlike <input> tags, <button>‘s can contain other html elements as their labels. <input type="button"> can only accept a string as its label text (css styles not withstanding).
Additionally, the <button> element accepts a wide range of uncommon but useful attributes regarding multiple forms and click actions. See the MDN page for more details.
As for one “out living” the other, the HTML standard is remarkably backwards compatible. Humanity will put men on Mars before either is eliminated from the HTML standard.