So, in the actual game Dad 'n Me, the major point of it is comboes - and that idea is being kept here. Whilst most characters have a "normal" attack button and a "special" attack button (I presume), this father and son duo actually have a difference in that they have a "weak" and "strong" button, more like in their original game. Interestingly, grabs can be performed as parts of combos and the like, and despite the vast combo system, there's probably going to be a couple more added for variety - including one simple-ish one that will let you grab an opponent and cargo-hold them a little like DK's grab.)
Simply tapping, say, the weak button twice will give a basic jab attack - but doing various different button combos between the weak and strong will result in various different combo attacks, with different effects. Given that the original game only has you play as the Son, however, there's some changes from the original game.
Whilst all of the combos are kept - yes, all of them! - you actually play as the Dad, though your attacks are much the same as in the original title. Your Son will sit on your shoulder, as in the original title screen, but I'll get to this in a moment. Basically, all of your attacks are close-up, and rely on you being in close-quarters to your opponent. As with other beat-em-ups, Dad n' Son are actually a mix between a rushdown, a bruiser, and a grappler - a heavyweight in the form of Dad, with one or two grabbing-based moves, but mostly based on getting up in your face and beating you down. Dad has surprisingly high ground speed when dashing, but to counter this, he has a very slow walk, as well as a low jump and very slow air speed, making him - much like Little Mac - very ground-centric. Not only this, though, but the large size of Dad makes it surprisingly easy to combo him, if you don't go in for a full-frontal assault.
I'm going to presume that, in this case, specials work a little like in Smash Bros. The original game was a little odd with how it handled "special" attacks, as it sort of mapped projectiles to the power-attack button, so I'm taking some creative liberty here.
"Neutral B" for the pair, well, isn't actually that. Instead of having a shield button, it has Dad "charge" his Son, at which point his Son dashes forward, and then "sits" at a distance from Dad based on the charge level (A little like Rosalina and Luma.) Son is interesting in this case, as whilst he has the same basic moveset as Dad, he does have some limitations. First off, he doesn't have any special attacks of his own - only his Dad is strong enough to carry around stuff for specials.
Also, whilst Dad's large size makes him a solid close-mid range character, simply due to his build, Son is somewhat smaller (though, still strong in his own right.) As a result, whilst Son deals less damage than Dad, as well as having less range due to being smaller, he also has less lag than Dad both before and after his attacks, which results in an interesting side effect - it's actually possible to run an attack string with Son by bufferring combos with Dad, due to Son's attacks coming out quicker. This could be used in a lot of interesting ways alone. Otherwise, however, Son moves at the same speed as Dad, and if you press Shield again, Son runs back (in a similar "dash" to him running away) and hops back on Dad's shoulder.
Whilst this means that yes - Dad n' Son actually lack any special attacks or projectiles - the pair of them more than make up for it with their exceptionally powerful close-up game and unusual mechanic of using the Son seperately to the Dad. Even so... there's something else to mention.
Son actually has his own healthbar seperate to that of Dad, which is exactly 60% of Dad's own starting health (So, for instance, if the game starts with everyone at 100HP, Son only has 60HP.) This, however, doesn't apply if Son is being carried by Dad. When carried, Son's health doesn't actually go down at all, even if they're both hit - meaning tanking hits as Dad and then sending your Son out at the last minute is a theoretically valid strategy. However, if Dad is K.O.d, Son is as well (Whilst this is for gameplay purposes, maybe we could presume it's because Son had a breakdown or something?)
Interestingly, they actually do have some minor differences too. For instance, Son has a knife that he uses for certain attacks for his special combos - whilst Dad actually uses the bloodied chainsaw that he has in the title screen, and on the back of his car in the intro sequence of the game.
On the other hand... Dad's own response to his Son being K.O.d is very different. In reference to the Rage mechanic from the original game, if Son is K.O.d but Dad is still on the screen, Dad will fly into an uncontrollable rage, which gives a variety of effects. As well as a purple smoke coming from him, Dad will slightly grow in size, appearing more muscular - as well as becoming slower as a result, both in terms of movement and in terms of lag, and having an even lower jump and a heavier weight. So, you might think this is a debuff, right? Bigger hitbox, less jump power, and doesn't even have the power of speed? Well...
Rage Mode massively increases one thing - the outright power of Dad's attacks. Whilst the attacks are slower and hence laggier, Dad at this point becomes the equivalent of a "miniboss" of sorts. Attacks from him are buffed in power significantly, and his slightly larger size means that he can reach even further than before.
Even so, however, Rage Mode is pretty balanced. Dad's much slower speed, larger hitbox, lower jump power, and laggier attacks offset the incredible increase in his range and outright power pretty well, and lighter, faster, or more projectile-heavy characters can quite easily take advantage of this.
As a result, this character ends up being a weird fusion of the puppeteer, rushdown, grappler and heavyweight archetypes. Whilst such a bizarre mix might not work in a lot of traditional fighting games simply because of how weird it is, I think this sort of bizarre fusion works well to represent Newgrounds as a whole - a bizarre mish-mash of unusual and unique things. I mean, how else could we have a stickman with cool pants, a mysterious and unsettling man with long fingers, and a soldier fight on a hand-drawn battlefield?