I have been reading Dan Jones' books about the Plantagenet kings and queens of England as well as watching various documentaries on Youtube since this is something I knew almost absolutely nothing about -- being an American from the public school system the extent of my knowledge was that William the Conqueror conquered England, King John signed the Magna Carta, Henry VIII killed all his wives, and Richard the Lionheart was a crusader.
Dan Jones and others cover the relationship between Henry II and his chief aide Thomas Beckett who was not from the nobility but who was basically made one of the most powerful men in the kingdom until he joined the church and the two had their big falling out.
Later, Jones covers Edward II. He suggests or implies that the problem Edward II faced was that Piers Gaveston was an outside made too powerful and too close to the king. The nobility grew to resent this relationship. (With the same being said for Despenser) However, it seemed to be a fairly similar relationship to Henry II and Beckett.
Jones addresses the possibility of their being some sexual dynamic to Edward II and Gaveston, but I am skeptical that this was the major cause for the strife during Edward II's reign. I understand that the allegations arose years later and were not contemporary, and it also seems strange that the greatest critics of the king did not have use his sexuality to attack him politically when had there been any substance to these claims.
TL;DR: What, if anything, was the major differences between Thomas Beckett, a commoner and favorite of the king with inordinate power compared to his class and Piers Gaveston?
I have to admit I'm less familiar with Edward II's reign, but I honestly don't see much comparison between Becket and Gaveston besides the obvious point of them being mentors of an English king who wind up murdered by nobles. Their careers and the circumstances of their deaths are quite different.
Becket wasn't of noble birth, as you note, but the idea of 'nobility' was (probably) still shoring up in the 12th century and he was definitely from the upper classes, receiving a Latin primary education already in England. He would have joined the Church when he studied in Paris, although he may just have been in minor orders (i.e. not yet ordained as a priest). Back in England, his career began in Church as Archdeacon of Canterbury along with serving as Lord Chancellor in the secular government. Here, the conflicts began basically when Becket was chosen Archbishop of Canterbury, coinciding with what we would call his "religious conversion" in the monastic sense, which means not changing religions but being much more religiously/spiritually focused and zealous. In this case, it also manifested in Becket's turn towards being a champion of the Church's rights and privileges--just as Henry was seeking to circumscribe the legal rights of the Church in his kingdom. Over these issues, then, the two of them came into conflict. They made inroads towards peace, but ultimately they were each committed to their courses of action. Whether or not Henry intended or wanted or sort of wanted his knights to murder Becket, the two were in open conflict at the time of his death.
In contrast, it seems to me (in my more limited knowledge here) that Gaveston remained a close confidante of Edward II throughout his princedom and kingship. His exiles and eventual murder came at the hands of nobles displeased with Gaveston's treatment of them and influence over the king. Where there was an underlying political issue, it concerned the relationship between king and top-shelf nobility, not secular versus Church authority.
Does that help clear things up?