I dont understand why to vary the weight. What situations call for a heavy/light nymph?
Heavy nymphs are probably used most. For fishing/rolling bottom.
Medium nymphs - for doing the same in shallower and/or slower water, you get a more natural drift than a weighted one. Or suspending in the water column, often during hatches. If using a team of nymphs, often you have a medium and a heavy, which gives you two different actions, and one rolls bottom while the other rides just above it. Can play with which is the point and which is the dropper. I'd classify almost all store bought nymphs as "medium". Meaning no wire underbody. But on heavy gauge hooks.
Light nymphs - for fishing high in the water column, or even nearly floating just under the film. Tie with a dry fly hooks. Avoid any wire "decoration". Grease as necessary.
This is usually hatch situations. Some bugs, for instance, sulphers, the nymph swims to the surface and hangs out there for a long time before emergence. As a nymph. They are of the swimmer variety of nymph, and they look like schools of minnows congregating on the surface. As the moment approaches to hatch, their swimming efficiency goes to heck and they start doing kind of a side stroke just under the film for a while, like a sick dying little minnow. Eventually they stop with their back to the surface, the back splits open, and out pops a dun onto the surface leaving a shuck behind. It is very common that you see boils or porpoises, but the fish are keying in on those fidgeting nymphs near the surface rather than winged duns on the surface. Emergers or no-hackle dries like comparaduns will take fish in this situation so long as you get a good portion of the body under the film rather than on it. But floating nymphs work better, IMO, if you can keep them near the surface. I usually fish a dry dropper in these situations. High floatin dun with a light nymph trailed by a foot or so, greased tippet just ahead of the dropper. If fish start taking the dun, snip the dropper off.
Other bugs don't transform this way. Some may emerge from their nymphal shucks on the bottom and "fly" to the surface as duns (traditional wet flies are deadly in these situations). Some may crawl out of the water to hatch. And even ones that do go to the surface as nymphs, other species may be much quicker about the hatching part, making duns more important.
Why couldn't i accomplish it with splitshot?
You can, but not as well. The biggest issue most have with deep nymphing is strike detection. And getting better is all about line control. Keeping a tight line, and staying "in touch" with your nymph. Having a big old shot between your rod tip and nymph puts you in touch with your split shot, not your nymph, and many strikes go undetected this way. The more and bigger the shot you have, the worse this problem gets. Weighting the nymph can help you get away from the use of shot, or, at the very least, allow you to get away with smaller or fewer shot.
Also, IMO, shot tend to snag much easier than weighted nymphs. And shot also imparts more drag.