Welcome to ECN Chester.
The problem with the 3-wire set-up is that the current-carrying neutral is also used as the protective grounding conductor.
That means that in normal use, if there should be a break or bad connection on that neutral, then the frame of the dryer could become energized at 120V via the control circuits.
Swapping the 3-wire receptacle for a 4-wire one while leaving only a 3-wire feed doesn't really accomplish much. If you had linked the neutral wire to both the neutral and ground connections of the new 4-w receptacle (which I believe is against code anyway), then all you've done is make the system 4-wire from the receptacle to the dryer. A bad neutral on the feed would still result in the same situation as described above.
Leaving the grounding terminal open on the new receptacle means that there's no protective ground on the dryer at all now.
