I'm looking into camera lenses and have seen two referenced as what certain bloggers use to take a certain kind of light-filled "perfectly overexposed" strong-bokeh picture that I am partial to
The lenses you mention have excellent reputations and price tags to match, but you can shoot photos in that same light-filled, low depth of field style with less expensive lenses. For example, Canon makes the EF 35mm f/2 that obviously has an even larger max aperture (and therefore blurrier foreground and background) than the EF 16-35 f/2.8, but the price is $600 instead of $1500.
Consider your first example. There's lots of light on the woman's back and on the background behind her. She's facing away from the light, but there's still plenty of light on her face, so there must be something outside the scene that's providing light. That could be a flash, but more likely it's a reflector or just a big light-colored wall that's bouncing some of that bright light back at her. The photographer made sure that her face was properly exposed, which let everything in direct sunlight be somewhat overexposed, and the aperture was selected to provide the desired amount of blur in the background. You can take that shot with a built-like-a-tank L lens that's outstanding in every way (weatherproof, too!), or you can take that shot with a more affordable lens that's still very good optically. But here's the thing: just using the L lens won't make your photos look like the example. The lighting is what's really important in getting that bright look that you're after.
if I'm going to be shooting at a large aperture (let's say f/2.8) on either lens, is there a noticeable difference between what the resulting images will be?
Well, you're talking about two different lenses with different apertures and focal lengths, so yes, there's going to be a noticeable difference between them. Longer focal length means narrower angle of view. Let's say you take a photo with each lens where the subject fills the frame. The shot with the wider lens (i.e. shorter focal length) will include more of the background.
Is it ultimately a question of just prime vs. zoom lens?
Prime lenses are simpler, with fewer elements, so they tend to be sharper and less expensive than zooms, and they tend to have larger apertures. But the differences you seem to be concerned with have more to do with focal length and aperture than anything else.