Does Drawsteel use Vancian spell slot magic rules like old editions of D&D and current Pathfinder do?
No. It's more similar to 4e except there are no daily powers. Characters have a set of abilities, some of which can be used at will and some of which require a heroic resource. Different classes have differently named heroic resources but they all work in similar ways: at the start of your turn you get either 2 or 1d3 of it (depending on class), and there are various triggers that give you more of it, usually to a max of 1/round. In addition, you start each encounter with 1 HR per Victory (previous encounters since you last took a respite (long rest, 24 hours or more in a safe place like an inn), which do not have to be a fighting encounter). You start with abilities costing 3 and 5 HR, and at higher levels you gain abilities costing more.
The only daily attrition in Draw Steel is Recoveries, which act as a sort of hit point reserve. You can spend one as a maneuver to heal 1/3 of your max stamina, or someone can use an action to let you do the same. There are also several class abilities that let people spend recoveries. If you're familiar with 4e, they're basically healing surges except a little more accessible.
One of the main tensions in the game, from a game POV, is that the longer you go the more easily you can access your cooler abilities (because you start each encounter with extra heroic resources), but your recoveries get spent. Taking a respite resets your recoveries, and converts your Victories to XP (16 XP = 1 level).
But Drawsteel decided to go for "lets compete head to head with what many suspect is the number 2 RPG, one which has a great company it's fans love, top noche customer support, an insanely regular product release cycle, a huge library of adventures, the best online tools in the industry, communities that make it super easy to fill tables or find a GM, and has almost if not actually no negative vibes about it, and gives fans of it's playstyle almost exactly what they want...
yeah lets take on those guys."
To me that choice just seems really baffling.
That does not seem to have been the thought process. It seems to have been more like "I liked a lot of things about 4e, and there are no games around that do those things. And it seems like an increasingly bad idea to put all one's eggs in the D&D basket. So let's make our own game that's the game we want to play rather than stuff for someone else's game."
So I'm focusing on Vancian Magic because really that's the only "lead weight" Pathfinder has. It's their 'weakpoint' that you could use to make an argument to pull people from one table to the other - but only if Drawsteel doesn't have the same weakness.
The main draw of Draw Steel seems to be that you get to be Pretty Damn Cool right from level 1. For example, a level 1 Fury can, as an at-will action, hit a target for 5/8/11 damage (depending on result), and push them 3/4/6 squares. Should they get knocked into something, they will take 4 damage + the remaining push distance. And that's without even using any resources, or including bonuses from their kit (which will likely either be a straight +2 to damage or +4 on just tier 3 results). As a 3-Ferocity ability, you could run through a line of enemies, hitting each of them for 2/3/5 damage (again, plus kit bonus), with potential extra damage to the last one if you took AoOs on the way..