kind of a broad question. The Nelson Funday is actually put on by the people that run the Nelson Ledges road course. There is NOT any in-car instruction provided and it's a much more open format.
Most days at Mid-Ohio are actually run by other groups (SCCA, NASA, BMWCCA, PCA...etc). Unless you were referring to the actual "Mid-Ohio School" which is still run very similarly to the other organizations with classroom instruction, in-car instruction combined with track time.
The biggest difference is the lack of the classroom/car instruction and the requirement to run around to those type of things all day. At a funday you merely pick a run group, go out, come back in and watch/ride with others.