Most of the small- to mid-sized, top ~50 private universities and LACs were founded before the Civil War.
Historically, these schools have tended to focus on liberal eduction (not technical/pre-professional training).
Many big state universities were founded after the Morrill Act of 1862, which granted land to states for colleges specializing in agriculture and the "mechanic arts” (hence the letters “A&M” in the names of some of these schools.) Their mission was to meet state and national needs for technical leaders during the second industrial revolution. Many of these schools have strong engineering programs, as well as pre-professional programs in fields such as architecture, agriculture, business, communications, and nursing (which many of the older, private “elite” schools still don’t cover at all).
As other posters have indicated, these distinctions aren’t always hard and fast anymore (if they ever were).
In addition to MIT, Caltech, and Harvey Mudd, other private schools with strong engineering programs include Stanford, Princeton, Penn, Columbia, Cornell, Northwestern, JHU, Duke, Rice, Vanderbilt, Carnegie Mellon, Olin, Rose-Hulman, Bucknell, and Cooper Union.