No, a datum and ellipsoid are not equivalent. For a loose definition, think of the ellipsoid as defining size and shape. The datum then fixes that ellipsoid to the earth.
NAD83 (various realizations) and WGS (another set of realizations) use almost the same ellipsoid GRS80/WGS84, and were originally designed in the 1980s to be equivalent. Since then, NAD83 has remained fixed to the North American plate, while WGS84 is...not. Thus the two datums are drifting further apart with time.
Think also about non-earth-center-earth-fixed datums. Many used the same ellipsoid, but the ellipsoid was 'fixed' to the earth at different locations. Datums that share the same ellipsoid could have a coordinate pair that was hundreds of meters apart on the ground. These older datums like NAD27 and ED50 have a fundamental or origin point. For NAD27, it's a point in Kansas called Meades Ranch. ED50 used the Helmert tower in Potsdam. The coordinates of other positions can be calculated from that origin point plus the ellipsoid definition.