There are two approaches to fertilization: dosing directly into the tank with liquid fertilizers (water column dosing approach) and using substrate based fertilizers (rootzone fertilization). Most plants can take in nutrients through both routes, and there are advantages to having nutrients in both locations.
This involves dosing liquid fertilizers directly into the water column. This involves measuring out a certain amount of liquid fertilizer and pouring it directly into the tank every couple of days. Its advantage is that it's very precise & consistent - we can measure down to the parts per million (ppm) of how much of a certain nutrient we want in the water and how regularly we add that amount. It's also easy to reset - by doing large water changes. It is also easier to measure nutrient content in the water column than in substrates.
Water column dosing has no reliance on plant root systems. This is especially useful when cultivating plants with damaged or immature root systems (such as freshly cut stems). For plants that are topped and replanted frequently, having accessible nutrients in the water column is helpful as their root systems need time to grow in.
The water column is open to all, so larger, faster more aggressive growing plants will take what they can, leaving less aggressive growing plants with less. Thus the disadvantage is that water column dosing must be done regularly and with consistency. It is not like substrate feeding where you can place a root tab and it dissolves slowly over a few months.
Plants that feed from the substrate will compete only directly with stuff that is planted in the immediate root zone. You can choose to feed specific bunches of plants by adding root tabs but not others.
Some elements are easily absorbed through foliage, such as potassium, so it makes sense to dose these through the water column. Certain ferts are more reactive (PO4, Fe) and may precipitate out of the water column after a period of time depending on your water chemistry. Reductive processes in the substrate zone make these more easily available to plants, so having these nutrients in the substrate zone makes sense.
Substrate fertilization involves either using nutrient rich soils (or aquasoils) or using root tabs. Using root tabs in inert substrate is less effective than using soil substrates as soil has the ability to bind with many mineral elements including ammonia, and hold it in a form that is easily accessible by plants whereas for inert substrates root tabs will just slowly leech into the water column.
Some species such as Centrolepis drummondiana (often called "Trithuria blood vomit") and Eriocaulon quinquangulare shown above grow significantly faster and more stable with root fertilization (especially with ammonia rich substrates) in a way that is hard to replicate just water column dosing.
Using a rich substrate - with leaner water column fertilization is one easy approach to running a planted tank, and hence the popularity of enriched aquasoils in this hobby. Using a rich substrate also means that one does not have to learn the troublesome approach of calibrating water column fertilizers as precisely - this makes it easier for beginners.
Some compounds such as ammonia are very beneficial for plant growth; plants preferentially uptake ammonia rather than nitrates for growth. However, dosing ammonia into the water column is not a good method - it makes it available to algae, and it is also quickly oxidized by nitrifying bacteria. Having ammonia in the substrate is a much better approach; ammonia binds to soils and is readily available to plant roots. It is also a much safer location as it is not available to algae. This is the main reason why many aquasoils are infused with ammonia - and why they work so well.
The main disadvantage of relying on root tabs is that there is low precision - how much nutrient does a root tab hold and how long exactly does it last? Does it release nutrients in a linear fashion ? Does it only work for plants in the immediate substrate zone? The answers are largely guesswork. Even nutrient rich planted tank substrates such as commercial aquasoils become less potent over time (6 - 8 months or so), as water soluble nutrients leech out.
Soils can sustain low tech tanks with slow growth rates for long periods without additional dosing, but most CO2 injected tanks have higher requirements. We can enrich the soil nutrient content using root tabs or adding fresh aquasoil.
Somewhere along the hobby came the myth that plants with large root systems were "root feeders" while stem plants with smaller root systems fed through their leaves.
While it is true that many species do very well with root fertilization, 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 preference to take in nutrients through the substrate or water. Many stem plants with smaller root systems also benefit significantly from rich substrates. Bucephalandra and Anubias species that can be grown without substrate, actually grow faster if they are allowed to root in nutrient rich soil.
Many species grow better, more stable in good soil. But this is specie dependent and not always related to the type of plant or whether it has a large root system. Most easier species can be grown purely with water column fertilization - this includes many common Swords, Crypts and Lotuses that will grow without special substrate arrangements.
Unusually large root systems (for a given species) indicate that the plant is starved of nutrients and is thus channeling energy into root growth and nutrient seeking functions. This is particularly obvious in floating plants, where it can be easily observed that there is significantly more root mass development for plants grown in nutrient deprived waters compared to plants grown in nutrient rich water.
The main reason folks think about pure water column dosing is to avoid using soils/aquasoils which may be messy or expensive. While it is certainly possible to grow great tanks with inert substrates, it requires much more sensitivity, knowledge and effort to manage the exact water parameters (unless you are just on planning on growing the easiest of plants) that works for a certain tank style than most folks realize.
If you are depending purely on water column dosing - you can't for example, drop NO3 levels steeply to get rid of GDA - your plants will starve. Pure water column dosing methods invariable ends up with a huge amount of monitoring and tweaking water column variables on the user's part. ( Not to mention disproportionate amount of time on forums searching for ways to tweak water chemistry) What time saved in managing soil is wasted on the above. Having a very rich water column also means existing algae outbreaks are more severe. And failure to manage algae issues is possibly the number 1 reason for folks quitting the hobby.
Depending purely on the substrate means that water soluble nutrients get depleted quickly with water changes. Not all plants may be rooted as well - modern day aquascapes make a lot of use of plants that grow on wood & rock. If you have 100% rooted plants and are diligent on keeping your substrate enriched, you can run planted tanks that are purely substrate fed.
Water column fertilization is a fast and effective method of supplementation. Plants have also shown to uptake certain elements such as potassium, very efficiently from the water column.
Using a rich substrate, while dosing some nutrients through the water column, a balance of two systems, is by far the easiest, more effective method of growing plants and avoiding algae. It is therefore best to introduce nutrients from both angles. This is also why every large scale successful commercial planted system out there is based on the two pronged approach - aquasoil matched with water column dosing.
Water column dosing is easy nowadays as many brands sell All-in-one mixes. We prefer APT complete (of course!) on an auto-doser, and insert osmocote balls into the aquasoil substrate selectively every month or so for certain plants that we want to grow faster. This allows the tanks to be on auto-pilot where fertilization is concerned!