My quick research tells me that recurve bows and their advantages have been known since antiquity, so why did the bowmen at say Agincourt use huge longbows instead of the faster, more powerful recurve designs?
How big are the advantages? The are two advantages to consider:
Stored energy. A typical Turkish-style reflex-recurve bow will store about 20% more energy than a longbow of the same draw weight and draw length. To match the energy, a longbow needs either a longer draw length or a higher draw weight. The difference results from the different force-draw curves - a reflex-recurve bow typically has a convex force-draw curve, while a longbow has a slightly concave force-draw curve. The stored energy is the area under the curve (i.e., the integral of F.ds over the draw). A long straight bow (like a longbow) has an almost straight (i.e., only slightly concave) force-draw curve, while a short straight bow has a more concave force-draw curve. Thus, a long longbow will store more energy for the same draw weight and draw length than a short straight bow. This is one reason why longbows are long. The other main reason also follows from these points about stored energy: more energy can be stored for a given draw weight if the draw length is longer. For a wooden self-bow, a longer draw length requires a longer bow, to avoid overstraining the wood (unless you do things like make paddle bows, which come with their own drawbacks in efficiency). Using a huge longbow lets you store almost as much energy (about 80-85% as much) as a reflex-recurve bow of the same draw weight and draw length.
Efficiency. The efficiency is the fraction of the stored energy that ends up as the kinetic energy of the arrow. This is dependent on the mass of the arrow, and longbows have poor efficiency for light arrows. A bow with heavy limbs gives a smaller fraction of its energy to lightweight arrows than a bow with light limbs, all else being the same. This is not an advantage of recurve bows per se, but an advantage of lightweight bows. However, since the previous point about stored energy drives longbows to large size, and large size results in heavy limbs (because they're large), longbows are less efficient with light arrows than lightweight reflex-recurve bows (and other lightweight bows). There is an easy solution to improve the efficiency: use heavy arrows. A longbow shooting 85-100g arrows is reasonably efficient (in comparison, a lightweight reflex-recurve (e.g., Ottoman) bow might shoot 30g arrows). The cost of using heavy arrows is that they move more slowly (for the same kinetic energy), and range is reduced.
Since longbows were used, there must be some advantage. The main advantage is cost. A longbow is cheap - it uses cheap materials, and takes relatively little labour to make, compared with a composite bow. Cheap matters! Archers-in-training need to have bows to practice with, and these will need to be replaced now and then. If the government dictates that all men will practice archery, archery needs to be affordable.
A further advantage is that longbows are resistant to weather conditions that can be bad for composite bows. The glues used on composite bows are not waterproof, and rain and humidity - both present in abundance in England - can weaken such bows (Babur, in his autobiographical Baburnama, notes the short working life of composite bows in wet conditions in India).
Large bows are also typically more damage resistant than lightweight bows. This can result in more reliable performance on the battlefield. Further, lightweight high-efficiency composite bows are not only more delicate, but the high strain of the strung bow (due the reflex) can result in unfortunate things happening like the bow turning inside-out if the limbs are twisted. A straight bow, on the other hand, is nowhere as severely affected by things such as twisted limbs.
Cheap, weather-resistant, more robust on the battlefield - all important advantages, and can balance the bow being only 80-85% as effective at storing energy.
Note that the second "disadvantage", needing heavy arrows for efficiency, was also chosen by some users of reflex-recurve bows. The Japanese yumi (a wood-bamboo laminated bow) was even longer than the typical English longbow, and suffers from the same efficiency problems. The Japanese adopted the same solution, heavy arrows, and their arrows could weigh more than 125g. The Manchu/Qing bow was a very large horn-and-sinew composite bow, close to the length to an English longbow. Again, heavy arrows were used (often about 125g). The use of large bows allowed longer draw lengths, and more stored energy for the same draw weight. Long-range performance suffered, but these bows excelled at short ranges.
If shooting at armoured targets, and trying to penetrate the armour, shooting at short range is desirable. At long ranges, arrow energy is lost through drag, and the likelihood of penetrating armour falls. If shooting will be at short range to try to get through armour, heavy arrows are not a serious disadvantage (thus the Japanese and Manchu choice of large bows and heavy arrows).
For measurements of stored energy and efficiency of reflex-recurve bows (Ottoman-style), see http://www.atarn.org/islamic/Performance/Performance_of_Turkish_bows.htm
For some measurements of simple longbows and their efficiency, see H. Pontzer et al., "Mechanics of archery among Hadza hunter-gatherers", Journal of Archaeological Science: Reports 16, 57-64 (2017), https://doi.org/10.1016/j.jasrep.2017.09.025
For the cost of longbows, see the response byu u/Hergrim to https://www.reddit.com/r/AskHistorians/comments/7qzfyk/how_did_a_english_longbow_cost/
In addition to what u/wotan_weevil greatly explained already, there is one important factor to be considered, to wit, the origin of these weapons. As already explained, recurve bows are not exactly 'more powerful' in terms of stored energy than the self-bows. They are simply more powerful than the self-bows of comparable size. Composite recurve bows were quite popular among Asiatic steppe nomads because the latter were most commonly moving, hunting and fighting on horseback, and thus required small-sized weapon that would be easy to maneuver while riding (including easy storing it in the saydak should the hand weapon had to be drawn quickly). A relatively small stature of the inhabitants of these regions also could have been a factor, although most likely a minor one. In other words, the size of, say, Mongolian bows was already severely limited by the conditions they were designed for and the invention of composite, recurve bows was simply the result of making the best of a bad situation - use of additional materials, such as sinew and horn (the one stores additional energy while stretching, the other is good at storing it during compression) and the specific construction (curved tips of the arms work a bit like pulleys allowing for more efficient use of energy) allowed to create small, yet considerably powerful weapon.
English longbows, traditionally very long, were already operating at the upper end of the human capacity, at least when we consider an average 15th-century English archer. Large infantry formations would have not benefited from smaller size of the weapon in any meaningful way, and there was little point to enhance the already high draw weight of the bow as this would render them nigh-inoperable. The complexity and the materials used made them less than useful in humid climate of northern Europe, as already explained, although the technology was definitely well-known, as some 15th century crossbows clearly show that the bows were made of wood, sinew and bone composite.