Some parts of this I’ve adapted from an earlier answer of mine.

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The Plantagenet dynasty self-destructed in the conflict between Lancaster and York, known today, but not at the time, as the Wars of the Roses. I haven’t read Philippa Gregory’s books but I’m thinking your confusion here may result from Richard of York’s self-identification with the name Plantagenet. In his bid for the crown he used the name politically to highlight his royal descent, in particular his possibly greater claim than that of Henry VI, the Lancastrian king. Before this time, the name Plantagenet was rarely used to denote the royal house, though the association was always there, especially in the beginning. So, let’s start in the 12th century…

Henry II became King in 1154 as the son of Henry I’s daughter Matilda. Henry II’s father was Geoffrey, count of Anjou (also of Maine and Touraine). Geoffrey excelled in battle and tournaments, and in both he wore a sprig of a broom plant (Old French: plante geneta) in his visor to identify himself. Plantagenet was something of a nickname, like Lionheart or Longsword, which became attached to Geoffrey and his son. Henry II was Henry Plantagenet, Henry of Anjou, Henry Fitz Empress (his mother had briefly been married to the HRE Emperor), and this is just a few of his monikers besides King. Historians refer to his immediate dynasty as the Angevins, while they still held Anjou. In 1205 Henry’s son, King John, lost Anjou and many other places to Philip II of France. From that time forward, historians use the name Plantagenet. I say historians, because monarchs didn’t use it so much. The name Plantagenet is a modern historical convenience and a medieval historical memory.

The dynasty then goes from John to Henry III to Edward I to Edward II and Edward III. More than two centuries with one line of father-to-son royal descent. Primogeniture meant people got very accustomed to knowing who God had ordained the next King should be. At the end of this cycle things get squirrely.

Edward III had lots of children, but for our purposes here I’ll mention 4 of his sons in descending order of age:

• Edward of Woodstock, Prince of Wales, known to history as the Black Prince

• Lionel of Antwerp, Duke of Clarence

• John of Gaunt, Duke of Lancaster

• Edmund of Langley, Duke of York

Edward the Black Prince died young, before his father. With England at war in France and several adult sons of Edward III thriving, you wouldn’t necessarily consider the Black Prince’s 7 year old son Richard as the best choice to take the throne, but that primogeniture thing meant that little Richard became Richard II under the regency of his uncle, John of Gaunt, Duke of Lancaster. Lionel, the oldest surviving brother, preferred books and art to warfare and politics. He had a daughter who married a man named Mortimer, creating the Mortimer line of Plantagenet descent. John of Gaunt was the most prominent and wealthiest man in England during his regency, but when little Richard got big he turned out to be the King nobody wanted. Richard II’s character is a rabbit hole unto itself, but suffice to say he managed to alienate his entire nobility, in particular his cousin, Henry of Bolingbroke, John of Gaunt’s eldest son. In 1399, After John of Gaunt’s death, Henry, now Duke of Lancaster, usurped Richard, with the nobility’s support, and became Henry IV, first of the Lancastrian line of kings.

Henry IV never sat comfortably on the throne. He faced uprisings and rebellions for the rest of his life. Yet, when his son Henry V won the battle of Agincourt, subdued France, and became a Netflix star, few in England were complaining about the Lancastrian ascendency. Unfortunately Henry died of illness at the height of his powers leaving an infant son to rule England (and France?) as Henry VI under a new regency council. By this time in the mid 15th century there was a vast network of related magnates, all with some degree of royal blood, all with opinions, desires, and grievances. During Henry’s long minority political unity was maintained among his many uncles, but powerful men operating for so long in the absence of a central authority, developing their own political / military followings (called affinities) planted the seeds of militant factionalism. As the uncles died off and Henry took control of the government, a new generation of nobility found themselves either with or against the court party surrounding the King.

Political opposition to the court party centered on Henry’s cousin Richard, Duke of York. York was descended, on his father’s side, from Edmund of Langley, John of Gaunt’s younger brother. But on his mother’s side, York was a Mortimer, descended from Lionel, John of Gaunt’s older brother. This gave him a powerful claim to the throne, senior to the Lancastrian line. York was one of the wealthiest landowners in England, second only to the King, and with a vast affinity behind him. His descent made him heir apparent in light of Henry and Margaret's un-fruitful marriage. York was also much better at doing "king stuff" than his soft-headed cousin. He was very effective as the king’s deputy in France, succeeding in a war the court party hoped to end. This made him a threat to the court party, in particular Edmund Beaufort, Duke of Somerset, who was another descendant of John of Gaunt (the Beauforts came from John's 3rd marriage) Both Somerset and York were related to the King in similar degrees; Somerset was among Henry’s chief advisers while York was politically marginalized. York was replaced in France by Somerset (who proceeded to screw everything up) and sent to govern Ireland, a crap assignment at which he actually excelled. Enmity betweed York and Somerset pushed political differences toward military conflict.

This is where you get Yorkists Vs. Lancastrians. Nobody was using the name Plantagenet until Richard decided to push his brithright and began referring to himself as Richard Plantagenet. Later Tudor historiography (and Shakespearean histories) portray the rise of the Tudors against the backdrop of a dying Plantagenet dynasty.

(Oh, and if you're looking to know where the Tudors come from, I've got this other answer which I hope explains that part of things reasonbly well.)

Sources:

John Gillingham - The Wars of the Roses

Dan Jones - The Plantagenets

Dan Jones - The Wars of the Roses

Alison Weir - The Wars of the Roses

I see why it can be confusing, they are all are related but then history books start referring to them as different families even though the claim to the throne is because they are the same family. There is a detailed answer in this thread on the history of the Plantagenets but I do think the poster slightly misinterpreted your question which I believe was more along the lines of, why do we refer to one side of the family (Yorks) as Plantagenet but not the Tudors on the other side. If it wasn't obvious this is more a historical label, not one used at the time. It also often misused as a means of suggesting illegitimacy towards the Tudors.

The label itself is simple enough to understand however, once you know the rules. Plantagenet refers to Henry II and all his direct descendants. So in terms of the Wars of the Roses this is all of Edward III's direct descendants. The Lancaster Kings from Henry IV down to Henry VI are all direct descendants as are the Kings and contenders of the House of York. Both Lancaster and York wings come from different sons of Edward III. The Tudor's are direct descendants of one of Edward III's sons, John of Gaunt, the same son of Edward III as the Lancaster (through John of Gaunt's second marriage). This is why they are considered part of the Lancaster House, but the descent of the Tudors is through a woman, Margaret Beaufort, great-granddaughter of John of Gaunt from this second marriage. I'm afraid traditionally children of marriages are considered of their husband's lineage so Margaret's son, Henry Tudor (the future Henry VII) is considered not to be a Plantagenet. You mentioned Elizabeth of York, she is considered a Plantagenet as she too was a direct descendant of Edward III but her son Henry VIII is not considered a Plantagenet as his fathers house is Tudor.

It is through this that Richard III is given the title "last of the Plantagenets" as an epitaph despite the only difference being that he descended from male heirs rather than a female.