The concept of a difference between Fly-By and Fly-Over predates RNAV, GPS, and even VOR/DME.
The most basic navigation happens without any navaids at all, and just a map, compass, airspeed, and stopwatch. Fly to a point defined by some visible feature, reach it, and turn. That's Fly-Over.
When you draw that route on a chart, in order to be precise you have to account for turn radius to determine the ground track you're flying after rolling out of the turn. (When I drew low-level navigation charts flying in the Air Force, we had templates that we used to account for the TAS & associated turn radius.) When the lines on the chart are drawn from point-to-point-to-point, then you have to do something other than flying over the waypoint and then turning to point at the next one... either you lead the turn so that you roll out on that point-to-point straight line course, or else you "S-turn" after flying over the point, to correct back to the that point-to-point line.
When the navigation has limited precision, the difference between the methods doesn't make enough difference to worry about... if my turn radius is a mile, I have 100 miles until my next waypoint, and my wind correction is based on a forecast from a couple hours ago, then the crosstrack error introduced by a 30 or 45 degree turn is probably less than the various other errors that I'll be accounting for. On the other hand, when I can see my crosstrack error to the 0.01 NM on a modern FMC and Navigation Display, and the autopilot is going to zero that out, it's really nice to NOT have it S-turning back to course.
Most waypoints these days are fly-by so that the charts depict all the airways and tracks as point-to-point, and we let the FMC sort out how far we need to lead the turn, given our True Airspeed right now. (And, of course, the FMC doesn't care if there's a VOR station sitting beneath the point or not.)
But there are times, especially when obstacles come into play, that the procedure designers want the aircraft to stay on the given track all the way to the defined point, and don't start an early turn (in order to prevent something undesirable from happening -- probably, in order to assure clearance from some obstacle or terrain feature).
The difference between the two types of waypoints matters more, now with tighter navigation tolerances and highly precise databases, but the fundamental conceptual difference has been around for a long time, probably almost as long as aircraft navigation & Mr Jeppesen's original charts.
For graphical depictions:
All graphics are mine; I'll edit the text into text as time allows.