Jumps are analog. That statement singlehandedly encompasses an incomprehensible amount of control that can be had. This is the true "20XX" of falcon. It is a tech skill that is hard to see, but the effects are easy to feel. No one will freak out when you master this, because it doesn't involve pressing buttons as fast as possible. Despite that, the amount of control you gain from this will make you win, and keep winning.
The mechanics of an analog jump apply to both the neutral jump and the dash jump, with minor differences. There are two major aspects to be considered in an analog jump. They are INITIAL VELOCITY (IV) and aerial mobility.
Let me focus first on the IV. IV is determined solely by the second to last frame of a characters velocity. Imagine the control stick as representing a scale from 0-10, including all non integers. The exact center of the stick is 0, and the furthest right is 10. Absolutely any number that you hold on this scale during the second to last frame of jumpsquat will exactly determine your IV. There is no deadzone for this mechanic. Of course, you can also hold backwards varying amounts, but thinking of 0-10 helps for simplification.
I want to stress that the last paragraph is quite likely the most important of all. I suggest re-reading it and trying to understand it fully.
Since this concept alone encompassed a ridiculous amount of options, I'll show my point by only showing 4 options. These are dash jumps with 4 different IVs (-10, 0, 5, 10) Sleepyk*
Anyways, aerial mobility works by either adding to or counteracting your IV. Thus, your IV and aerial mobility are meant to work together in harmony to achieve your goal. Using them together in a masterful way, falcon can jump into many different ranges with many different angles. As far as I can tell, all airborne frames contain the same potential for aerial mobility. Thus, there is one critical frame before being airborne that determines IV, and the rest of the frames before you land are simply used to add to or work against that IV.
The major difference between neutral jump and dash jump is that if you dash jump with maximum IV, aerial mobility in the direction you jumped does nothing. You are already traveling at max velocity, so you simply can not hold forward and travel further. Falcon will not slow down if your control stick is resting at 0 while airborne. Acting against your velocity will have a drastic effect, of course. Furthermore, if your IV was equal to a random number below 10, such as 5, you would indeed be able to hold forward after your jump. Falcon will approach max velocity once more, but never surpass it.
In a neutral jump, falcon will lose velocity of his own accord. If your IV is 10 and you hold 0 while airborne, your jump goes about half distance. You must continue to hold 10 the whole time while airborne if you wish to reach the maximum distance. Overall, neutral jump gives you significantly more control over your velocity than dash jump. This is why I think it's usually better to use as an attack when you're already within that range.