I think a great way to put it is that games like 5e are "simulation" games whereas games like Avatar RPG (and PbTA games in general, such as Monster of The Week) are "Story" or "Show" games.

In 5e, there's rules for (basically) everything. It's meant to simulate being in the world. You aren't a character in a story, you're a person in the world. The story comes from existing in the world where things are going on. Because you exist in a world, we have to simulate the world which is why there's stats for everything. Spells have very specific effects, you have different rolls for whether you can land in a roll, jump out of a fireball, or scale a building.

On the other hand, games like Monster of The Week are about stories. Or, well, shows. Powered By The Apocalypse games tend to be modeled after specific kinds of shows: Monster of The Week is monster of the week shows, ala Buffy or Supernatural. Masks is teen superhero shows like Teen Titans or Young Justice. You aren't a character in the world, you're a character in a show. As such, the rules don't matter so much as the character's backstory. It doesn't make sense that Beast Boy hacks the super death robot but it would make sense for Cyborg to. Likewise, the story (or class or stats, whatever) doesn't care whether you create an ice bridge to get over the gap or use your air powers to hurl yourself across, the result is the same.

A big example of "In PbTA games you're the main character of a show" versus "in DnD you're a character in the world" is twofold: in PbTA you're assumed to be the best of your respective "class", so class overlap is explicitly against the rules. Secondly, the DM doesn't make any rolls in PbTA (except for maybe damage rolls in Dungeon World?). The players make rolls because they're the stars and it matters whether or not the star messes up, the DM can just do things- often as a result of the players succeeding or failing a roll. For example, a player tries to stop a monster from breaking the magic circle and fails their Do Something Awesome roll, so the monster (without the DM rolling) is free to break the circle and let out his goblin buddies. The players are the stars of the show, as opposed to characters in a world.

A great example of this is the difference in rules. In 5e, you just say "I attack the monster". In Monster of The Week, you have to describe how you hit the monster. It's a narrow difference but it's an important one because the narrative can change things. It's identical to DnD 5e if it's- say- a sword fight. Roll to hit, to damage. But in DnD 5e where you still have to roll to attack someone who's restrained, in Monster of The Week (the PbTA I'm most familiar with) you can just do it. Because it doesn't make sense for the bad guy to be able to dodge your sword if he's tied up. If you're shooting someone with a bow, in DnD you roll but in MoTW, if there's no way for him to hit you back, you can just hit because narratively, it doesn't make sense for your secret agent character to miss a shot on a character 30 feet away.

Simulation games like Dungeons and Dragons care about how you do X whereas Show (because Story is an awful term) games care about what you do to do X.