Some have told me the f/2.8 lens requires more glass to achieve f/2.8 and therefore it lets in more light and can shoot at a faster shutter speed at the same aperture - but I really don't think that makes sense.
Basically incorrect. The entrance pupil for 300mm at f/8 will be 37.5mm in diameter, regardless of the diameter of the lens' front element. So you won't gain any shutter speed advantage by using the f/2.8 lens at f/8 as compared to using the f/4 lens. When you're stopped down to f/8, the light striking the parts of the front elements of either the 150mm diameter f/2.8 or 75mm diameter f/4 lens beyond the 37.5mm diameter circle in the center of the element is not being allowed to pass through the aperture diaphragm.
You will get an advantage with regards to the amount of light used for focusing and composition as most cameras meter and focus with the lens wide open and only stop the lens down to the specified aperture setting a split-second before the shutter opens. This will very likely mean faster speed and more precision in terms of autofocus performance. Exactly how much difference will depend on the capabilities of the camera's AF system.
The only other possibility that makes sense is that typically lens manufacturers will produce a better build quality on a lens that can achieve f/2.8, so, for example, you might get sharper images because of better build quality, but that has nothing to do with the lens' ability to stop to f/2.8.
That's pretty much it in terms of image quality. There is an additional IQ consideration, though, that does have to do with the lens' ability to open up to f/2.8.
The typically larger front element of the f/2.8 lens mean that when stopped down to f/8 the aberrations introduced on the edges of the larger objective are blocked by the aperture blades. Since these aberrations increase in severity with an increase in the diameter of the front element when shot at maximum aperture most telephoto f/2.8 lenses have more extensive aberration correction than narrower aperture lenses of the same focal length. It is theoretically possible to make an uncorrected f/2.8 telephoto lens that would have poor image quality and would be outperformed by a good f/4 lens, but the market doesn't demand such a lens. Almost all constant aperture f/2.8 telephoto zoom lenses and f/2.8 telephoto prime lenses are of very good optical quality because the market demands it.
In general, with long telephoto lenses the larger aperture models will slightly outperform their smaller aperture counterparts when both are from the same manufacturer and the same generation of lens design. But there are no real "consumer grade" first party lenses in the prime supertelephoto category with which to compare them. They're all very well corrected. (There are much cheaper third party fixed narrower aperture mirror lenses with lower optical quality and very cheap fixed narrow aperture refractive lenses with abysmal optical performance because there are few, if any aberration corrections in the design of the lens.)
With wider angle lenses it is usually the case that optical quality compromises must be made to allow wider apertures at such wide angles of view if field curvature is to be corrected to a flatter field of focus. It is much easier to correct a narrower aperture wide angle lens for field curvature than to correct a wider aperture wide angle lens for field curvature.
Depending on their intended usage, some wide aperture prime lenses in the 50-85mm focal length range do not correct for field curvature. When tested at a single focus distance optimized for the center using a flat test chart as a target these lenses appear to be very soft on the edges and in the corners. But often they are very sharp on the edges and in the corners, just not at the same focus distance as the center of the field of view because the shape of the sharpest focus at any point in the frame is part of a sphere.
Lenses corrected for field curvature to give them a flatter field of focus (often called plane of focus though that is somewhat of a misnomer for all but a pinhole camera - no lens is perfectly corrected for field curvature) will do much better across the entire field when tested using a flat target at a single focus distance. Such well corrected lenses tend to produce bokeh that is harsher or 'busier' than lenses less corrected for field curvature. Thus, a lens like the EF 85mm f/1.2L II which is not well corrected for field curvature has a distinctively smooth bokeh on the edges of the frame, but is totally unsuitable for flat field work, such as document/art reproduction.