WHAT IS IT, REALLY?
Most aquarists use the term "hardwater" generically to refer to water that is high in mineral content - this can mean either the water being high KH or high GH or both. It is common for "hardwater" to have both high GH and KH because sources of hardwater are areas with limestone [CaCO3] - which contributes to both GH [Ca ion] and KH [CO3 ion].
However, hardwater can be high in GH and low in KH and vice versa.
Minerals that can raise GH without raising KH are minerals that contain calcium/magnesium but do not contain carbonates : Magnesium sulphate [MgSO4], Calcium sulphate [CaSO4]
Minerals that can raise KH without raising GH are minerals that contain carbonates but do not contain calcium/magnesium: Potassium Carbonate [KCO3], Sodium bicarbonate [NaHCO3]
WHAT IS IT, REALLY?
As with the term hardwater, aquarists use the term softwater generically to refer to water with low mineral content. However, we should really define the variables precisely - whether the livestock/plant is sensitive to high KH or high GH values as these two are independent.
plants that prefer 'softwater'
Most "softwater" plant species are actually sensitive to high KH values and not GH. It is more accurate to then say that they prefer water with low alkalinity [low KH]. Most plants are tolerant over a wide range of GH unlike KH.
Typical softwater species such as Syngonanthus , Rotala macrandra, certain Eriocaulons, can grow well in high GH waters as long as KH is kept low; about 3dKH or less. Many aquarists may be less limited in their plant selection than they think, if they have high GH but low KH water. Between 1-2 dKH you can keep sensitive softwater species (Syngonanthus, sensitive Eriocaulons, Rotala macrandra, Tonina, Ludwigia pantanal, Trithuria blood vomit etc.). Between 2-8 dKH you can keep 97% of all commercial aquatic plants in optimal condition. 8-12+ dKH you can probably grow 95% of species well, but some will be sub-optimal. Above 18 dKH or so, more issues start arising with regards to growing plants.
The red Eriocaulon, Eriocaulon quinquangulare prefers very softwater. (less than 3dKH)
Syngonanthus species tend to favor very softwater and acidic substrates. (less than 3dKH). Ludwigia glandulosa (purple plant above) is much more flexible and takes medium-hard water well.
Trithuria sp. 'Blood vomit' prefers very softwater, while the purple plant above, Bucephalandra 'Brownie ghost' actually does well in hardwater. When mixing species, one must give way to softwater plants.
Rotala macrandra (mini type 4 shown here) prefers softwater water, but can tolerate KH values up to 7dKH or so.
what about fish?
Generally fish are also more tolerant of a wide range of GH compared to KH. Most commonly available species of fish can live in a surprisingly large range of both GH & KH. However, certain specific species, and especially their eggs and fry, may require low GH along with low KH water to breed well or display their full coloration. Important to have about 4dGH if keeping shrimp.