Was it the act of acknowledging children that would make them "official"? Or was it just that if the relationship was not hidden socially? Or did they get some official title? If so, is there euphemism involved? (Like "friend of the king" or "adviser to the king" or something?)
How does this interact with the expectation of lifelong monogamy? Famously, Henry VIII couldn't divorce and remarry and centuries later Edward VIII couldn't marry a divorcee. Why is having an official mistress OK but divorce and remarriage means you lose your throne, even in the 20th century?
‘Quasi-official' is perhaps more accurate. The king's maîtresse en titre would have a noble title, an apartment at Versailles, and a pension, just like all the other ladies at court; the king would openly spend time with her. But it wasn't an official title like 'groom of the stool' or 'first lady in waiting'. Sometimes the king would be weedled into acknowledging by rescript the bastards he had sired (without mentioning the mother's name and without including them in the legitimate line of succession).
Madame de Maintenon, the most famous mistress of Louis XIV, was not really a mistress at all - she was apparently too pious to have sex with him until they were properly married. (Spinsters, take heart: Madame de Maintenon was 38 when she snagged Europe's most eligible royal sex hound - who was actually younger than her.) The marriage was never officially acknowledged but lasted for three decades.
As for your second question, difference is that Henry and Edward wanted to be married, not just have sex. Edward suggested a morganatic marriage - but hardly anyone believed in the divine right of kings by the 1930s and he was forced to abdicate.