Henry the Young King, as he is usually called, had no power at all. He asked his father many times for castles and lands, but didn't get much. He also asked for legislative powers and never got any. His position was purely ceremonial - he was a king without a kingdom. He spent most of his days competing in tournaments but in 1173 he harnessed general discontent at the reign of Henry II to rebel and claim actual power, but the revolt failed and all he got out of it was a bit more pocket money, which he spent expanding his tournament team. In 1183 he rebelled again but died of dysentery on the march at the age of 28.

Edward V, who I must admit I am less knowledgable about, died at the age of 12 in the Tower of London. On the 14th April 1483 his father Edward IV died, making him the new king. Richard, Duke of Gloucester and Edward V's uncle, was to be "protector" and people wanted Edward V crowned asap to prevent a regency and the awkward politics they often entailed. Instead, Richard used his position to delay any coronation. Richard also had Edward's adult relatives and staff arrested and then executed after dinner with him on April 29th, making him the next adult in line to the throne. Edward V and his younger brother Richard were then put in the Tower of London. Their sister Elizabeth was ignored because she was a woman (this would come back to bite him). On June 22nd it was claimed that Edward IV's marriage to Edward V's mother, Elizabeth Woodville (thanks /u/frumperbell), was illegitimate and that Edward and his brother were therefore ineligible to be king. On June 25th Parliament met, agreed, and appointed Richard as the new king, Richard III. The princes remained in the Tower of London and were last seen around July/August of 1483; it is assumed that they were eventually murdered.

In Henry the Young King you have someone who was actually crowned, "reigned" for years, and attempted to actually rule twice. In Edward V you have a boy, potentially excluded from succession, who wasn't even crowned and was murdered as a boy. If anything Henry deserves the numeral and Edward doesn't. The difference is in how subsequent generations viewed them.

Henry the Young King was a nice guy but there wasn't much reason to remember him. At the time of his death he was widely revered as "the father of chivalry" and is credited with developing the tournament into an enduring sport, but in terms of chivalry and valour he was overshadowed by his tournament captain William Marshall. Where Henry the Young King never held any actual power, Marshall died as regent of Britain, played a key role in stabilising the kingdom, saved it from a French invasion, and helped create Magna Carta. When Marshall died it was Marshall who became "the greatest knight" and the embodiment of chivalry; the world had forgotten Henry the Young King.

Edward V, despite doing nothing, is remembered because of the manner in which he was deposed and the reputation of Richard III under the Tudors. By contemporary accounts and surviving documents, Richard III was a good ruler who greatly improved legal representation, clamped down on fraud, translated all laws into English, and encouraged the printing of books. However, when the Tudors took power they embarked on a propaganda campaign against Richard III and the Plantagenet dynasty to make themselves look more legitimate, and the treacherous way Richard seized power played a large part of that. This was important because Henry Tudor's claim to the throne was very dubious and had promised to marry Elizabeth, Edward V's sister, to boost his credentials. The worse Richard III looked, and the more Edward V looked like a king, the easier it was to claim that Elizabeth and by extension Henry Tudor was the rightful ruler. This was hugely successful - in William Shakespeare's Richard III Edward is portrayed as a king, wise beyond his years and fully capable of seeing what Richard III was planning, whilst Richard III is a bloodthirsty tyrant. That image endures to this day, and an enormous part of what makes Richard III bad in the eyes of the public is that he murdered the old king. So even though Edward V was never actually king and might not have even been eligible to have the title, it suited later generations to write about him as if he was, including giving him that 'V' at the end of his name.

So, to summarise, Henry the Young King doesn't get a numeral after his name because history quickly forgot him and Edward V does get the numeral because it suited the Tudors and made its way into Shakespeare.