From the 16th century onward, in France they had a specific status. The younger son (or daughter) of a king would be a "Son/Daughter of France", their child would be a "Grandson/daughter of France" and their descendant "Prince/Princess of the Blood". (note this only applies to legitimate descendants). This meant that they were indeed treated as superior to other nobles. They were considered "Pairs de France" (Peers of France) a specific status which meant they had the highest status in court after the direct royal family.

During the times of the Direct Capetians, the official status of younger Capetian sons was no more exalted than whatever title they were granted. This includes such men as Robert of Burgundy, Hugues of Vermandois, Robert of Dreux, Pierre of Courtenay, Robert of Artois, Alphonse of Poitiers, Charles of Anjou (ancestor of the Capetian House of Anjou), Robert of Clermont (ancestor of the Bourbons), Charles of Valois, and Louis of Évreux. In practice, however, being the sons and brothers of kings gave them unofficial levels of clout and often greater wealth than non-royal counts. Charles of Anjou and Charles of Valois, for example, were able to muster the resources try to claim foreign crowns, though only the former succeeded. Charles of Anjou's descendants, who became kings of Naples and Hungary, enjoyed generally close ties with their French cousins: Charles of Valois married his double second cousin, Marguerite of Naples, Countess of Anjou; Louis X's second wife was Clementia of Hungary; and Louis I of Anjou was designated as heir by Jeanne I of Naples, which subsequent Valois, including Louis II of Anjou, Rene of Anjou, Charles VIII of France, and Louis XII of France, became obsessed with making good on.

However, again, this wasn't officially granted precedence. This is best illustrated when Isabelle of France supposedly protested her son, Edward, giving homage to her cousin, Philippe VI of France, because Edward was the son of a king while Philippe was only the son of a count — nevermind that the count had been her own paternal uncle, Charles of Valois.

The accession of the Valois kings, of which the aforementioned Philippe VI was the first, saw the creation of the title prince du sang, though agnatic descendants they wouldn't been given official ranking over all other members of the peerage until 1576.

How the other cadet branches other than Capetian House of Anjou fared differed: the Vermandois went extinct within three generations and the Courtenays' fortunes were ruined by their adventures in Byzantium, after which half of the family that went east took up residence at the court of Charles of Anjou and married into his family and the few that stayed in France declined into genteel poverty and political irrelevance. In the 17th and 18th centuries, the Courtenays tried to claim the status of princes du sang, but without success. However, the Dreux and Bourbons did very well for themselves, all things considered: the last member of the House of Dreux was Anne of Bretagne, the twice queen of France, and the Bourbons eventually got to become kings of France and later, Spain and the Two Sicilies.

I'll do bastard sons in another comment.

After the start of the High Middle Ages, bastard sons of kings held only as much status as their fathers' felt like granting them. Succession of bastard sons to royal and noble titles was far from uncommon in the early Middle Ages (see: William the Conqueror or Ramiro I of Aragon) but it declined rapidly after the mid-11th century, except in Wales and Scandinavia. A number of French and English medieval kings most often arranged for acknowledged bastard sons to marry heiresses or get high-ranking positions in the church.

Examples of the former are Robert of Gloucester, Philippe Hurepel of Boulogne, William Longespée of Salisbury, and Richard de Chilham and an example of the latter are Geoffrey, Archbishop of York. Some royal bastards, such as Philippe Hurepel; Joan, Lady of Wales; Marguerite de Belleville; and the Beauforts; were legitimated by the Pope, though often with the proviso that they were barred from royal succession.

Sometimes, bastard siblings and their legitimate siblings were close; for instance, the unfortunate legitimate sons of Louis I of Orléans seem to have adored their bastard half-brother, Jean de Dunois, who loved them in return and kept watch of the family possessions while his half-brothers were being held captive in England. On the other hand, relations between Geoffrey, Archbishop of York, and his legitimate half-brothers were sometimes tense, though that's not too surprising since relations between his four legitimate brothers put the fun in dysfunctional.