I've read a lot of books by Bernard Cornwell (the Grail Quest novels), and in those books he goes great lengths to ensure historical accuracy. At the end of every book he talks about what he made up for the sake of the story, and what he wrote which was true to history.
In his books he heaps praise on the longbow, telling that it could more often than not punch through mail and gambeson, and even plate armour in the right circumstances. However, when I discuss the bow with other medieval warfare buffs many of them completely disagree, and some even go as far to say that "the longbow was nothing special compared to other bows". They conclude that it absolutely could not pierce steel plate armour, and only if the bowman was extremely close, accurate, and lucky could it pierce a quality mail shirt with a padded jack underneath.
When I read about some of the major battles of the 100 years war (Crecy, Poitiers, Agincourt) it is clear that while the bows were excellent for disrupting cavalry charges, the more prevalent deciding factors were the terrain, weather, and bad French leadership that gave England the victories. Also note that while the longbows brought down the French horses, they did have to charge uphill, and in some cases over pits and caltrops and in the mud which would have reduced the speed by a significant amount.
So is it true that the English longbow is nothing special? Would the English have fared any worse had they equipped their archers with shorter, lighter bows in place of the specifically tall, thick, yew bows? After all, an arrow will pierce any unarmored man, but if the longbow is just the same as any other bow in regards to armour penetration then what really makes it so special? Of course, it had long range, but I've read that so did crossbows, with some late period crossbows being able to outdistance them. I know that might sound a little too r/HistoricalWhatIf-y but it's more to illustrate the point that perhaps the longbows praise is unfairly given when other bows could have gotten the job done almost or just as well.
Unfortunately there's no way to systematically prove how effective 14th century longbows were. For one thing none survive to the present day and even if they did how could we test them? We can't exactly dress people up in medieval plate and have them run at archers who shoot arrows at them. That said my research is making some attempt to answer this.
One big thing to start with is the terminology we use when talking about the 'longbow.' What is meant by short bow versus a longbow? Composite bows are short, do we include them with short bow or are we only talking about self bows (i.e. bows made of a single piece of wood). What is the cut off for a longbow, is it 5 feet or 6 feet or more? I'm not trying to poke at problems in your question here, this is something lots of books discussing longbows don't really cover. The girth of a longbow is in many ways more important than its length as bows derive much of their power from their width not their length. Length helps but not as much as you might expect. The Traditional Bowyers Bible, a work about making wooden bows, includes a great chapter on the mechanics of archery that's a great introduction to the subject.
In more direct answer to your question there have been several experiments done testing replica longbows against armour stand ins. Peter Jones did two separate experiments and the Defense Academy did a round of tests attempting to update Jones' original methodology. There have also been several tests done (including some by the Royal Armouries in England) just to measure the velocity of longbow arrows (among other weapons). None of these tests are perfect and they all have their problems so I wouldn't take anything they say as gospel. That said, when you combine their results with Alan Williams' study of medieval armour (specifically the average thickness of full plate across different places on the body) it suggests that longbows couldn't really penetrate breast plates or helmets, where armour was the thickest, but could penetrate armour on the extremeties. Now this is all done in pretty optimal testing conditions for the longbow so bear in mind on the field this was way less likely and these places are areas where the arrow is far more likely to wound or incapacitate than kill. This isn't damning evidence but it's the best we have so far, in my opinion. When it comes to mail...things get even more complicated and I'm on a lot less firmer ground when discussing it so I'll mostly leave it aside for the moment.
For the use of the longbow in battle there are several opinions. For my money I agree with Kelly DeVries view which is best expressed in his article Catapults Are Not Atomic Bombs which is largely about reconsidering the effectiveness of pre-modern weaponry and includes a section on the longbow. DeVries argues that the longbow wasn't a lethal weapon but was more of a strategic one. By firing volleys of arrows at the enemy the English could disrupt enemy formations and interfere with their ability to draw up proper disciplined battle lines. You have to remember that longbow arrows are pretty big things and even if they don't kill or injure if you're standing there being shot with them it's certainly not pleasant. DeVries essentially includes the Longbow as one of the tools the English had, along with preparing the battle field with ditches or caltrops and the establishment of strong defensive positions, which enabled them to get a great advantage before their lines clashed with the enemy. His idea isn't perfect but I think it's the closest to correct I've read so far. The evidence from the Hundred Years War suggests that the longbow archers played a large role in the English successes but what exactly it is they brought to the army is up for debate.
On Longbow vs. Crossbow...well its complicated. The core idea is that crossbows were more powerful but took longer to reload. This isn't without its problems since not all crossbows were the same and they came in varying powers plus while longbows fired faster we don't know what the average number of arrows an archer could expect to have on a campaign was and that would severely limit his rate of fire.
Jonathan Davies looked at the inventory of longbows and arrows from Henry VIII's ships (a bit later than the 14th century admittedly but sadly lots of our longbow evidence is from Henry VIII) and calculated the average arrows per archer. If you assume something like 6 shots per minute (a figure reported by Robert Hardy and others prone to praise the longbows power) the average archer ran out of arrows in something like 5 minutes. The question then is, is it reasonable to assume that an archer would only contribute arrows for 5 minutes out of a multi-hour battle?
I hope that answers some of your question at least. My area of expertise is primarily the technology of longbows and crossbows with less of an emphasis on how they were used but I've still done my fair share of research on that too. I can try and answer any follow up questions you might have.
References:
Bourke, Paul, and David Whetham, “A Report of the Findings of the Defence Academy Warbow Trials,” Arms and Armour 4;1 (2007). 53-75.
Davies, Jonathan, “Military Archery and the Inventory of King Henry VIII” The Journal of the Society of Archer-Antiquities 44, (2001). 31-38
DeVries, Kelly, “Catapults Are Not Atomic Bombs: Towards a Redefinition of 'Effectiveness' in Premodern Military Technology,” War in History 4;4 (1997). 454-470
Jones, Peter N., “The Metallography and Relative Effectiveness of Arrowheads and Armor During the Middle Ages,” Materials Characterization 29 (1992). 111-117.
Jones, Peter N., 'A short history of the attack of armour', Metallurgist and Materials Technologist, 16;5 (1984). p. 247-50.
Richardson, Thom, “Ballistic Testing of Historical Weapons,” Royal Armouries Yearbook 3 (1998). 50-52.