These two fall under the category of water polishers, capturing small particles that are invisible to the eye. Both help in enhancing water clarity and adsorbing organic molecules.
Activated carbon is used to absorb odor and tannins and adsorb larger molecules well - these usually mean organic compounds with large molecular chains (such as medicine). It does not remove most fertilizers in a planted tank, contrary to popular belief as the molecules are too small to be captured. The fertilizer that it can remove is chelated iron, which is a large molecule consisting of iron (Fe) attached to an chelating agent such as an organic acid in Ferrous Gluconate. However, if dosing is done regularly the impact of activated carbon is insignificant - so you can use if you want. The list of what activated carbon removes effectively is located at the bottom of the page.
Is activated carbon necessary in most planted tanks ? No, but it can be useful in setups to capture small organic waste particles or tannins.
Purigen is a good alternative as it can be recharged (with bleach) and reused. It polishes water to very high clarity by capturing small particles too coarse to be captured by the filter.
ADA gallery in Niigata, Japan frequently use activated carbon to remove organic waste and tannins, especially in newer start-ups. These techniques have produced some of the most successful, stable, long term tanks in the industry - clearly destroying the myth that carbon is detrimental to running a planted tank.
These are not necessary in a planted tank and do not help solve algae issues. Plants require a certain amount of phosphates and nitrogen to grow well and these resins risk stripping the water entirely of necessary nutrients. If your tap water is within limits suitable for human consumption, generally there will be not such a large imbalance of elements as to cause serious issues.
Here's a table that gives you a good idea of what is removed by carbon.
High to moderate adsorption on:
Arsenic, Bleach, Chlorine, Colors, Dyes, Hydrogen Peroxide, Insecticides, Monochloramine, Odors (usually larger organic molecules), Detergents, Dissolved Organic Compounds (DOC), Hydrogen Sulfide, Mercury, Soap, Solvent
Fair adsorption on:
Complexed trace elements (due to organic chelator), Iron(as FE 3+), Lead, Vanadium.
Low to no adsorption on:
Alkalinity, Ammonia, Carbon Dioxide, Nitrates, Phosphates, Potassium