I'm fairly sure the phrase chop and drop applies to both. Some weeds can come back from surprisingly small pieces of roots and so are more easily killed if you remove the whole plant. Some plants will die and leave abundant organic matter in the soil if you cut the tops. I think deciding which approach to take is a matter of personal taste. Some people even encourage the perpetual resprouters as reliable sources of fresh organic material.
I like to leave the roots in the soil, except in the case of perpetual resprouters. In fact, in our area (and I'm also in the Texas Hill Country) a thick layer of mulch to protect the soil from the summer sun can do as much for soil health as nearly any amount of amendments. If you're short on potential mulching materials then the added root mass on the surface may actually be more beneficial.
The benefit of chop and drop is that you leave the root mass below the surface of the soil where it will continue to feed the soil food web. While the plant is alive, it is pumping root exudates into the soil to feed the fungi and microbial life. Once its dead, it still provides food, albeit a different kind of food feeding a different spectrum of microbial life.
In areas where frost and cold kill the plants, this "pulse" would happen on an annual basis and leave not only the root mass below the ground, but also a bio-mass mulch above the soil. Where animals graze, the root mass generally remains (unless they root it out, the way pigs can). So in general, chop and drop is more beneficial for building soil and feeding soil life.
Having said that, as you mentioned, many plants quickly re-sprout from the roots or the remaining stem of the plant. However, plants expend a great deal of their stored energy to do this. If you chop them off several times, it tends to either put them into dormancy or will kill them.
Mowing is the mechanical means of chop and dropping --- and blowing the seeds all over. If you can chop and drop before the plant goes to seed, it makes it easier to control what comes up next growing season.
Yanking a plant out, roots and all, creates disturbance that isn't in the best interest of your soil. I still do it, particularly in areas where I've mulched aggressively, as the weed comes out easily and I've got all sorts of other things growing there to pump life down into the soil. But if you can, it's better to chop them off at the soil surface and pile the biomass at the base of some other plant or tree.
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I chop-n-drop with a sharp hoe. Having spend many years primarily hand pulling weeds I would never go back. I can cover 10 to 20 times the area with the hoe in the same time, and don't end up with back ache afterwards.
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Excellent question and you have gotten some good answers already. The really big question is; are these plants something that can not live there because of what you want to do in that space?
Chop and Drop is a great avenue that will build soil just as nature builds soil, by plants dying and rotting and coming back from the roots or starting over from seeds dropped.
Pulling up root and all means you are redistributing some minerals to the surface, just like what happens when a tree is blown over by wind.
Cutting the top of the plant and letting it rot while the root remains allows the top part to rot and the root to either survive and regrow or die, this depends on the nature of the plant.
Disturbance is not bad for soil, in fact it is the way Nature works to create soil. Many people seem to think that Nature does not Disturb soil, but in reality nature disturbs soil all the time.
Fires sweep through (disturbance), floods wash down hill (disturbance) Winds uproot trees (disturbance) Volcanoes erupt (disturbance). You get the idea, nature makes her living by creating disturbances, then she starts to rebuild.
Trees blowing over or getting burnt up re-distribute minerals, bringing what was underground to the surface where it can nurture new growth. Disturbance is how nature builds soil. Only humans seem to think they know better than nature.
Animals are also part of nature's method of disturbance, their hooves trample vegetation into the soil, they mow grasses and browse trees (raising the lower branches which can prevent diseases from spreading easily), they poop and pee to provide fertilizer to the ground.
All these are soil building on a grand scale, nature's scale.
What you first need to do is make the decision, do you want rather sterile conditions that require you to make additions every year? Or, do you want to just cut, let rot in place and plant through what is already doing the work of soil building for you?
Either is acceptable, however, one of the principles of the permaculture ideology is to mimic nature, that is the best method for healing our earth mother.
To not follow the way nature works, we have to spend money or time or both trying to do what she does without our meddling ways. The traditional farmer takes the meddling ways to the extreme, the opposite of what we want to do.
Tilling mimics nature when a tree falls or even an entire portion of a forest is blown over. Tilling every year does not mimic nature, it is instead a state of constant disturbance, something which nature doesn't do.
No-till is a great method for traditional farmers, since it teaches them that they don't have to till the soil every time they want to plant a crop. That is where the idea of no-till came from, those of us teaching farmers how to get better crops with less expenditures of money.
No-till really isn't necessary for small plots of land, but it doesn't mean you want to dig up your garden every time you plant either.
Tilling is disturbance that is done deliberately so we can let nature start over, with the idea of our own intentions being the dominating theme.
To mimic nature we first have to fully and objectively watch how nature works her magic and then we need to understand progression of species, which is the tool of reestablishment, it occurs with or without people being around.
Have fun, do as little work as needed to get to your end goals for food, and remember, nature was here long before us, and it will be here long after the last one of us is gone.
We just need to learn how to stay out of her way and not poison her anymore.
By just doing that, we can help her heal. If we want to help her heal and grow food for ourselves, then we only need to mimic her methods.
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When it comes to my gardens, I have various weeds with various methods of dealing with them depending on their growth pattern, their seeding potential, and, quite frankly, whether I like them or not for other reasons in that particular location or in general.
I tend to like dandelions a lot (for food, and for worm/microbe habitat, and because I don't feel that they compete), so I try to work with them, letting the grow big and for several years, unless they are in an area that I want to grow another tap rooted plant like carrots which they do seem to compete with. I do sometimes chop them to let another plant have more light, like in my garlic patch in the spring when the garlic is small.
This is not at all the case for quack grass, timothy grass, (actually any grass), thistles, daisies, chickweed, and hedge nettle. These I have zero tolerance on. They spread via roots and tend to seed heavily if allowed. If I see them, and I have the moment to spare, they come out with as much roots as I can take-hopefully all of the roots. Generally I will mulch over the mess that I've made, since the exposed soil from the upturning as the plant was pulled is bound to have a lot of weed seeds (which perpetuates the problem). I have tried to chop and drop some of those tenacious spreaders, but they tend to take advantage of my weekness/laziness and come back to haunt me, and in short order.
My only success with chopping and dropping those types of plants, though, was to chop them down, plant potatoes and mulch heavily. The daisies and hedge nettles and chickweed couldn't take that. I still weeded out the grasses and thistles; they are the worst of my problems, but I knew that and weeded them before I chopped the rest.
There are others, like plantain, wild mustard and lambs quarters that I leave because I like to have them around for food and medicine. I might still chop and drop them, but I never pull them unless I think they are totally in the way, and then only if I am putting in seeds or transplants. The plant goes right where it grew, if it's a plant I like, otherwise it goes out of the garden, for compost, hugulkultur nutrients, or mulch under dense trees.
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