Somewhere along the hobby came the belief that plants with large root systems were "root feeders" while stem plants with smaller root systems fed through their leaves. A widely circulated thinking is that "Swords & cryptocorynes are root feeders and require root fertilisation".
Many plants with large root systems have large root systems because they grow in fast flowing or seasonal waters and the large root systems serves as anchors to resist being washed away, for other plants roots serve as a storage system. There is little correlation between the size of a plant's root system and whether is prefers to uptake nutrients through the substrate or water column. While Swords and Crypts do well with root fertilization, easier sub-species can grow well just with water column fertilization.
Most aquatic plants have adaptations that enable better absorption of nutrients through their leaves. Submerged leaves often lack the external protective tissues in land plants to limit water loss. The epidermal (outermost) layer usually shows very little sign of cuticle formation. Surface cells are able to absorb water, nutrients and dissolved gases directly from the surrounding water while the internal system of tubes (xylem) which normally transports water & nutrients from the roots to all parts of the plant is often greatly reduced.
Nutrients should also not be clustered as a general whole - the preferred pathway for potassium uptake may be different from the preferred pathway of nitrogen uptake for example. To this end, most plants seem to uptake potassium through their leaves efficiently. However, some plants may prefer N uptake through the form of ammonia through the root-zone rather than say through NO3 in the water column.
Most easier plants in the hobby can be grown without a nutrient substrate as long as there are enough nutrients in the water column ( this includes most species of hardy swords & crypts). Pickier species may do better with root fertilization, where certain nutrients may be assimilated more easily (PO4, NH3, Fe etc). Alternanthera reineckii, Eriocaulons species are examples where they grow more stable and faster with nutrient rich soils than purely with water column feeding. Pogostemon helferi, which can be grown on bare rock with water column dosing, similarly grows much faster when rooted in soil. This effect is also true for many plants that are not normally grown in substrate at all such as Anubias and Bucephalandra species.
That being said; root feeding often works well for many plants because we can introduce a higher concentration of nutrients at the root zone via root tabs. This is useful since most hobbyists aren't comfortable with dosing extremely high level of nutrients into the water column. We may not want to dose ammonia (which is very effective as a nitrogen source for plants) into the water column, but containing it in the substrate works well. Those that do sufficient water column dosing quickly realize that many common "root feeders" such as swords and crypts can grow well even purely with water column dosing. Substrates can also hold nutrients in an available form for plants through their cation exchange capacity (CEC); deeper substrate with slightly anaerobic conditions can reduce Fe3+ into the more usable Fe2+ format. These are the many advantages of root fertilization.
Lythraceae (Rotala, Ammania fall under this) species have shown to be sensitive to high water column fertilization when water KH is high and are thus more easily grown with rich root fertilization and lean water column. (Vin, 2019 AGA presentation has the details). For these species using rich soil is by far the easiest way to grow them if you do not have very soft water. Many Rotala macrandra varies, as well as Pogostemon erectus can be grown well in moderately hard water (KH 6 -10) using this approach.
Read this to learn about choosing between root or water fertilization.
Read this section for detailed plant care guides.