So both of your examples come from the 15th century, and this makes me happy. Because the heart of your question is "what's the difference between a king/kingdom and a prince/principality," and, well, medieval Western Europeans kinda spent the Middle Ages muddling blindly into a distinction. You can see (one tiny sliver of) the messiness of medieval geopolitics in the language of the sources: Geoffrey of Monmouth has no trouble crafting the dux bellorum of Pseudo-Nennius into rex Arthur; in 12th century German chronicles, basically anyone who holds land is called princeps right up to the emperor.

But by the fifteenth century, a larger bureaucracy and institutions like some kind of parliament, along with a standing army and the ability to maintain it (i.e. the right to raise taxes) were becoming more aligned with kings than princes. There were more informal markers, too, like a thriving central court for political wheeling and dealing (and to foster loyalty among one's nobles), a chivalric order (to foster loyalty among one's nobles), and whatever form of relationship with the papacy would ensure more say in the appointment of bishops and potentially abbots to powerful, ecclesiastical-lordship benefices (to ensure loyalty among one's ecclesiastical nobles).

But also, in legislation and chronicles, there begins to be an acknowledgement of the responsibility of the king as far as management of the kingdom. The king, with his advisers, had to balance the complicated relationship among the wants and needs of royal government, the nobles, public opinion, and this newfangled idea called "the public good" (I know, right? What is that even, asks the twelfth century lord). We know princes had started to expect this, because they complain when it's not working.

Don't take this picture of acknowledged centralized authority too neatly. In England, the conflict between York and Lancaster helped define high-level politics for most of the 15th century. And in France, the Duke of Burgundy not only had the king's son assassinated, he even got the support of traditional royal ally the University of Paris on his side. But in England, the battle ultimately turned out to be for the throne; in France, the duke had the assassination ruled a tyrannicide. A distinction of royal authority was real, recognized, and meaningful.

To examine why nearly all the high nobles of Europe really could, in fact, wait to be king, especially if they were going to be at war with the royal army anyway (welcome to France, or rather, Frances)--and see what they thought it would require to make that a reality--let's compare two dukes of Burgundy. Philip the Good (d. 1419-1467) made no movements towards a crown; Charles the Bold actively negotiated for one.

Philip was clearly determined to make Burgundy greater than it ever had been, and he wielded his principality like a sovereign polity to get it. Geographic expansion was his goal; marriage alliances, independent military actions, and negotiation were his major tools. In contrast to the later idea that the state (kingdom) controls foreign policy, Philip's armies went to war against those of England and the Holy Roman Emperor. With France divided by loyalty to the English interlopers or to the French holdouts during the Hundred Years' War, Philip oversaw a tidy (yeah right) switching of sides.

On the informal measures, too, Philip pushed to make Burgundy an international force in its own right. Marina Belozerskaya points out that despite modern fascination with THE Italian Renaissance, 15th century contemporaries almost all acknowledged Burgundy, and especially the Burgundian court, as the center of European arts and culture. Philip even established a chivalric order for his duchy. An independent identity for a powerful and in many ways independent Burgundy was his goal and his achievement, but never to the point of reaching for a crown.

So what was Philip getting out of sticking by France (whichever French crown that meant at a given moment)? Two of the crucial factors are pointed out in the murder of the Duke of Orleans (1409, on the order of Philip's father) that I mentioned above. Internally: politics was brutal. Hostile parties had spies and allies everywhere, with court offering a central place for interaction, intrigue, and danger. A claim that couldn't depend on the full support of everyone in the vicinity (plus anyone who might have an army to invade) was hazardous to one's health.

Externally: submission to a kingdom gave the prince an active and legal say in the policies of the kingdom. In cases like agitating for lower taxes, of course, it's kind of a wash since independence would mean no centralized taxes to pay. But in terms of foreign alliances, defense, internal alliances, convincing the king to tax your rivals more than you, dealing with wayward cities and robber knights, and other fun tasks of medieval power, a presence at the royal court and a spot in the Reichstag or in the Estates General of that well-funded kingdom next door/over your head would seem very advantageous.

Philip's successor Charles (r. 1467-1477), though--Charles had plans. From the moment he gained the dukedom, or possibly even before, he was in motion with a goal. He oversaw the organization of the northern Burgundian provinces into an Estates General type of body, generally the prerogative of a kingdom; he treated with the king of France to secure an independent judiciary for his territories and founded a sort of appellate court to prove it was justifiable; he established a permanent, standing army and the bureaucracy to support it.

And in 1473, Charles finally made his move. He turned to the Holy Roman Emperor to ask for a crown.

The funny thing is, Charles doesn't seem to have been as concerned about what the crown or title was. One of his reasons for looking to Frederick was the options: King of the Romans (which he probably knew was never going to happen), King of Frisia, or, why not, King of Burgundy. On its face, the lack of concern over what king title seems to suggest that personal honor was the goal. However, Charles' actions leading up to the request show it was more than that. Burgundy had to be a kingdom not just in function but in what was coming to be seen as form, if he were going to be king.

And more to the point, with the slowly converging ideas of king and kingdom, Charles believed Burgundy needed to be a kingdom to ensure it could continue acting in its own best interest. The best efforts of the Burgundian dukes in the early/mid 1460s had not worked to roll back what they considered French offenses to their polity (taxes, evil ministers, the works).

Frederick, of course, had other plans as well. He hemmed and hawed and stalled on Charles, but made sure to arrange a neat little marriage between his son Maximilian and Charles' daughter Mary. Charles' daughter, that is, who stood in line to inherit Burgundy after her father's death.

Although Frederick's plan was pretty clear to get Habsburg control over Burgundy, not see an independent polity on his northwest, the shambles that Burgundy was left in after Charles' death reflect that there was apparently one more requirement for a kingdom in the fifteenth century. One of the big problems with a heritable monarchy is what happens if there is no heir, or if there is reason to dispute the heir. This caused numerous problems across the fifteenth century, but the sovereign polities like Castile survived the turmoil. After Charles was killed in battle in 1477, Burgundy disintegrated. The southern provinces were absorbed back into France. The more urbanized and centralized northern provinces stuck together--Mary and hence Maximilian inherited and ruled them according to plan--but its nobles and city councils were able to extract far more concessions out of the new rulers than Charles had allowed. Philip had expanded and Charles had consolidated, but the government institutions, the bureaucracy and its officers, the confidence and recognition of its authority, could not hold the same thrall absent a widely-acclaimed male lineage heir.. Burgundy was certainly still a cultural and political entity, but it was securely a principality instead of a kingdom.

Recommended reading:

• John Watts, The Making of Polities: Europe 1300-1500, is the Cambridge Medieval Textbooks entry on the evolution of European civil authority in the late Middle Ages. That's where I'd start--it will be fire-hose dense, but it's reasonably readable, it's affordable, it doesn't assume massive piles of background knowledge of medieval political thought (...), and the bibliography will point you to anything else you might want. Including, yes, massive piles of background knowledge on medieval political thought.
• I talked about Burgundy here, but if you're interested in the Holy Roman Empire, the opening chapters of Thomas Brady, German Histories in the Age of Reformations present the more German side of things, interwoven with the story of how the Church was attempted to become a polity and...struggling.

The Republic of Florence had been a sovereign state for close to five hundred years by the time Cosimo Medici rolls around. What Cosimo did was, through his control of the state (and, importantly, his turbulent but successful relationship with the army) remove all disloyal administrators/functionaries in the Republic; these jobs had been the purview of members of noble families who could oppose him. Once he had consolidated himself in Tuscany, he then very delicately lobbied both Pope Pius V and the Emperor Ferdinand to have himself proclaimed Grand Duke. He initially wanted the title of King, but backpedaled once he understood the implications; in theory, the once crowned Holy Roman Emperor, the German Kaiser was also King of Italy, so if Cosimo proclaimed himself King others might interpreted his action as an intent to challenge Ferdinand for hegemony on the Italian peninsula. Indeed, this could be read as a legitimacy issue: other rulers would not legitimize a King, but a Grand Duke would be more acceptable, since the "Grand" title implies independence, but "Duke" doesn't imply sovereignty. Of course, it was never specified if the Grand Duchy of Tuscany was a subject of the Empire (as it would be given its history) or a subject of the Pope (seeing as Pope Pius was the one who crowned Cosimo); Cosimo didn't bother to clarify, he was looking to legitimize Medici rule over Tuscany, and that's what he achieved.

Italian rulers often ran into situations where they had all aspects of sovereignty apart from that irksome "Legitimacy." Gian Galeazzo Visconti, for example, paid a metric ton of gold to Emperor Wenceslaus to be proclaimed "Grand Duke of Lombardy" in 1360. The Visconti had ruled Lombardy for some two hundred years and had never bothered with titles, until Gian Galeazzo began pursuing a match with the daughter of the King of France, Isabel de Valois. Here, the ruler of Lombardy is literately paying the Emperor for recognition (and legitimization) of his rule over his domain; and should troublesome nobles challenge his rule, he could (in theory) call upon the help of the Emperor. No one was going to challenge Gian Galeazzo anyway, but I suppose this is an example of a thing a ruler had to do to "Not Get Laughed At." Like Cosimo, Gian Galeazzo avoided taking the title of "King," this time because by appealing to the German Kaiser he is tacitly acknowledging that there already is a King of Italy.

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Since /u/sunagainstgold focused on Burgundy post-Philip, I'll focus more on Burgundy's role in the Hundred Years' War and why it was never truly independent in the war that Philip's son wanted.

Burgundy's an interesting one and it dovetails into a much wider conflict. The Hundred Years' War, though it has entered popular culture as primarily a long-running war between England and France, was much, much more complicated than that. To give you an idea: at the time of the famous Agincourt campaign, there were no fewer than three major factions actively attempting to secure the French throne: the English (and their Gascon/Flemish allies and mercenaries), the Burgundians (under John the Fearless), and the Armagnacs (nominally under Charles of Orleans). The Bretons were also heavily involved in this conflict as they attempted to re-assert the independence they had lost.

The issue with anyone proclaiming themselves the King of France is that, by the time of this war, "France" had a clear meaning. Brittany, Burgundy, and English-held France were all legally subordinate to the King of France. All of these factions had previously recognized that fact. Brittany when it was conquered, Burgundy was always understood to be part of France given that it was awarded as an apanage [grant of land] to the king's youngest son, and England had previously recognized certain parts of their French holdings as subordinate to the King of France -- like Gascony, which Edward I paid homage for in return for it not being seized.

Thus everyone had agreed what constituted "France," and they felt that Burgundy, the territories of the Armagnacs, and the English holdings on the continent were part of France. Brittany was a bit weird in that it had previously been an independent kingdom and was attempting to reassert that.

All of which is to say that Philip the Good knew he was part of the Kingdom of France and everyone else knew it, too. It would be one thing for him to claim the title of King of France -- since his father (John the Fearless) had been murdered by the Armagnacs at a parlay, he certainly could justify such a declaration. That would put him at odds with the English (who also claimed the title, and at the start of Philip's reign probably had the best claim of all), but he could do it. But Philip couldn't just turn around and say, "I'm King of Dijon now" because Burgundy was recognized as part of France. He'd have to win independence at treaty negotiations after the war (which may well have been Burgundy's aim in initially aligning itself with England).

Ranulph Fiennes' Agincourt deals pretty heavily with the Hundred Years' War, the dynastic politics that precipitated it, and covers some of the Armagnac-Burgundian civil war. I highly recommend it. Despite the name, it devotes relatively little time to the Agincourt campaign.