The finer logo is rendering out right now, but I can show you the distinct difference that's very apparent once you see it.
I would've just subdivided the model to create more vertexes, therefore making the model itself more high quality and rounded out. But the issue with Blender is that every face needs to be in quads, having 4 vertexes. Unlike other things like video games, where they use triangles. When it comes to subdivision anyways.
The first version I had taken a screenshot of the logo, cleaned it up all to one color and making the rest transparent, converted it into an SVG and imported that into Blender as a 2D object. I extruded it in order to give it depth, and the entire process yields a result such as this:
So, there's not much I could do about that. I was starting to just smooth it out manually when it occurred to me that I'm an idiot.
I didn't just trace it out initially since I usually subdivide my models for a higher quality end result, but I seemed to forget that I don't have to do that. And in this case it looks just fine without being subdivided.
And all I had to do was more or less trace it out and keep it smooth.
TL;DR: OmegaFacepalm