They are great at purifying the water by taking in nitrates and ammonia as a source of nitrogen. By absorbing excess nutrients and reducing the amount of light, they often help control algae, especially in outdoor ponds. They also provide cover for shy fish.
In botanical terms, the major advantage they have over submerged aquatic plants is their access to air (specifically CO2). Approximately 50% of plant mass comes from carbon (through carbon dioxide) and while the concentration of carbon dioxide in air is around 400ppm (parts per million), underwater it is around 3ppm. This is why injecting CO2 makes such a big difference to plant growth. Without CO2 injection, it is akin to trying to build muscle without easy access to protein: possible, but so much harder.
Having access to air and a comparatively huge amount of carbon dioxide make floaters extremely powerful. So powerful that often they pose a danger to other plants when kept together in an indoor aquarium.
Their dominant position (closer to light, access to CO2) means that comparatively, they propagate much faster and quickly shade submerged plants. Most aquarium lights are not that powerful compared to natural sunlight - hence even small amounts of shading by floaters can cause the aquatic plants growing beneath them to suffer from a lack of light.
If you are keeping floaters in your aquarium, there is a good chance the submerged plants are not doing so well- simply because they don't have enough access to light. Even a little shading means a great deal.
In addition, if with good intentions you are providing a nutritious fertiliser, up to 90% of this added nutrition is likely to be absorbed by the floaters, leaving little (or nothing) for the submerged plants. Like animals, plants compete for nutrition, and have no mercy. The dominant species (and floaters will dominate submerged plants easily) will hoard all the good stuff.
Above: Amazon frogbit (Limnobium laevigatum) is popular in indoor aquariums as it has smaller leaves than its larger cousins such as Eichhornia crassipes (water hyacinth), Pista stratiotes (water lettuce) and Ludwigia sedioides (Mosaic flower) etc.
But without them I get algae!
Floaters are beneficial as they help absorb ammonia/nitrates and other impurities such as heavy metals. So in a tank with an insufficiently sized filter (or too much fish or light), they can often help reduce algae indeed by absorbing the excess waste in the water.
So we are not against floaters. However we don't generally keep them in tanks where we want the submerged plants to grow well. It is akin to saying it is possible but not easy to keep cats and birds in the same room.
Some popular smaller size floaters to use in a tank include Limnobium laevigatum (Amazon Frogbit), Phyllanthus fluitans (red root floaters- above), Salvinia natans, Azolla carolinana etc. Lemna minor (Duckweed) propagates a bit too quickly and is very small in size; which makes it difficult to manage for many hobbyists.
So what can I do?
If you want your submerged plants to grow better, but also want to keep your floaters, you can consider the following:
1.Confine the floaters to the edges of the tank, or an area where they do not shade your submerged plants. If you make this switch and have already weak submerged plants, it might trigger a temporary growth of algae. The solution is to add more submerged plants so that most of the area exposed to light is covered by plants. Open patches without plants and exposed to light are always vulnerable to algae.
2. Be vigilant in controlling the mass / volume of floaters and remove excess plants regularly. Remember that growth is exponential. By keeping your floater population under tight control, more light and nutrition would be available to the submerged plants.
3. Provide rich aquasoil and/or root fertilisers to your (rooted) submerged plants. Such nutrition would be less immediately accessible to the floaters, giving the submerged plants a better chance at securing vital minerals.