I can't really say one's better than the other, they're designed for slightly different roles as has been mentioned.
Here are some stats comparing longbows with asiatic recurves (like those the mongols and other nomadic groups used)
English warbow:
70 kg draw weight (average? 130-140 lbs I think; range is 80-180 lbs appr.), 182 cm string
65 gram arrow, 200 fps (feet per second) initial velocity
single target direct shot: 60m, 190fps
Max effective range with war arrows, lofted shot: 250 meters, 180 fps
max range with flight arrows, 280m, 160 fps
(note: it is possible to achieve flight arrow results of about up to 400 meters; I think the maximum range refers to the lethality of the arrow)
With heavier war arrows, english bows have a longer range with that specific kind of arrow, designed to try to pierce mail and plate armors of the time (1300s-1500s). the english warbow was designed mainly for infantry use in a relatively wet climate (bowmen could easily unstring their bow and keep their string dry in wet weather; recurves are more difficult to do this), used in large numbers in concentrated volleys into enemy formations.
Strategically, warbow archers were used in mainly defensive tactics. Archers would be placed on the flanks of a formation, with the men-at-arms and lesser experienced archers and crossbowmen in the center of formation. This way, the better archers would provide enfilading shots to an approaching enemy. Archers would loose volleys at long range lobbed up into approximate ranges, and then when the enemy got closer, they'd aim target-style. Often, in order to cicumnavigate the difficulties of armor (in general based on tests, except at very close range, the best armor at the time could defend from the arrows being lethal; penetration is still possible but whether it is lethal is questionable; this being said, a very small percentage of those fighting had access to the best armor of the time), english archers would often aim at the enemy horses to dismount their riders and disrupt a cavalry charge (pile of horses becomes an obstacle very quickly).
So,
pros:
-better concentrated volley (infantry is better for this as it is static) at longer range
-excellent defensive weapon
-uses war arrow capable of piercing plate at closer range, certainly capable of causing serious wounds at any range
-the weapon itself is easy to make, as it's a self stave
-wet climate and rain? not a problem (the finish on the bow wood solves that, both bow and arrows have canvas bags or cases, and bow strings can be kept under your hat, put there in seconds)
cons:
-less range with flight arrows,
-slower cast of arrows, less efficient weapon
-lacks mobility of horse archery
-slower rate of shooting than a recurve bow of a certain method,
Asiatic Recurve (note: this is a heavy recurve, they also used lighter bows for faster rate of shooting on horseback):
62 kg draw weight (range is probably similar to english warbow, this average is about 110 or 120 lbs. I think), 121 cm string
40 gram (war?) arrow, 220 fps initial velocity
single target direct shot, 50 m, 210 fps
max effective range war arrows, lofted shot: 250 meters, 200 fps
max range with flight arrows: 320m, 160 fps
(note: it is possible to shoot flight arrows much farther, up to about 500 meters)
Asiatic recurves were composite types of horn, wood, sinew, etc. glued together with fish glue. Mongol bows were well adapted to cool, dry climates, and were ill suited in warm, wet climates like southern China (one small reason why it took so long for the Mongol/Yuan dynasty to conquer the Song). Too much moisture, and the glue will loosen and the bow could be damaged. To prevent this, recurves are usually kept strung (stringing and unstringing them while in use takes longer to do) in leather cases until needed.
With slightly lighter war arrows, mongol bows can still pack quite a punch up close, though only later in their history (the golden horde mongol/tartar/russians who used these kinds of bow, mainly) would they actually encounter european plate armor. In the 1200s the main armors to deal with were mail and lammellar (small metal plates sewn together, providing an excellent arrow defense). Mongol archers (most of their army was cavalry, and most of these horsemen could ideally use a bow) usually carried two bows with them and up to 60 arrows of various kinds. They had a heavy bow (mentioned above) for armor piercing and long range, and they had a light bow for rapid rate of shooting (see the "Lajos Kassai" method of horse archery as an example of what I mean)
Strategically, the Mongols used their archers mainly on horseback, and for very mobile tactics. It is described in a treatise about the mongols and mamluks fighting that the mongols would attack in waves of horse archers, each firing whistling type, harassing arrows at about 300 meters or so to weaken and psychologically damage the ranks of the enemy; then at closer range, they would use war arrows within target shot range (50 meters, as close as 30) to finish the opponents off, then retreat and let other units join the fight. Then their "heavy" cavalry would close in with melee weapons if the opportunity presented itself. They also used the so-called "Parthian" shot while retreating, as well as many other tactics.
pros:
-highly mobile platform on which to shoot, horses
-versatile direction of shooting (you can shoot backwards while retreating)
-you can move into range, shoot, and out of range before a static force can respond to you
-great range with flight/light arrows,
-smooth release with thumb ring
-diverse selection of arrows, including whistling (the sound intimidates horses)
-efficient weapon with a very fast cast, resulting in a fast arrow!
cons:
-the composite bow is technically much more difficult and time consuming to make
-wet weather? better have that bow in the case!
-slower to string and unstring
-does not allow for as concentrated volleys of fire (due to horse archers needing to be more spread out, and them being moving platforms)
-extremely difficult to master with any reasonable accuracy or strength (Mongols had to train from early childhood in riding and shooting); more difficult than infantry archery because it's more complex
-a horse archer is a larger target and easier to hit than an infantry archer, and is more vulnerable to missiles.
Some Sources I took this information from:
Codex Guide for the Medieval Baltic, by J.H. Chandler (for sale at Drivethrurpg);
http://coursesa.matrix.msu.edu/~fisher/hst372/readings/amitai-preiss.html(article found here concerning mongols vs. mamluks, with differing accounts of how both sides fought)
I hope what I rambled about makes sense and is mostly accurate, feel free to tell me if it isn't.
