I agree with Dan, more or less (though we disagree slightly on the effects of types of arrows against types of armoured targets).
I believe that the obvious effectiveness of English/Welsh archers from c.1330-c.1440 was down to two main factors:
The latter point is incredibly important - in lands under English rule it was law to practice archery for males between the ages of 16 and 60, at least once per week, from Edward 1's reign onwards (it was only repealed under James I, I believe).
If you look at the latter stages of the Hundred Years War, in fact you will see that the French, Burgundians, Bretons and Flemish were using large numbers of their own longbowmen (the Burgunians even copied English deployment with wooden stakes and pits).
The idea that the English had longbowmen and the 'French' had crossbowmen is in part a myth - the English used crossbowmen as well (they are listed on muster rolls) and the 'French' also had archers - in large numbers by the 1450's. The French force that Talbot slaughtered just before Castillon was made up mostly of French archers...
However what is notable is that the 'French' never had as much success with longbowmen as the English did, despite also having them. What was the main difference? Well surely the weapon and arrows were much the same in design, but a critical difference was the law. Englishmen were raised in the bow, with sports like golf, football, skittles and cockfighting being made illegal! This law was repeatedly enforced under Edward II, Edward III, Richard II, Henry IV, Henry V and Henry VI.... This legal enforcement of archery almost certainly resulted in far more effective archers.
I find it ironic that some modern archers seem to see the English warbow as a proletarian item, a symbol of the success of the working classes over the aristocracy... when in fact the success of the English warbow is almost certainly largely a result of English monarchical and government policy and the oppression of national sports (like football).