Edward's children are a prince and princess, just like Andrew's, however, the parents decided to use the day to day titles Viscount Severn and Lady Louise as if they were "just" the children of the Earl of Wessex rather than have the burden of using HRH Prince James and Princess Louise.
Children get their titles from their father (yes, sexist, whodathunkit) which is why Anne's children are untitled - her husband refused a title on marriage to give the children a more normal life. If Anne were male, her children would be a Prince and Princess but, again, may be known as something else.
Harry's kids will become Prince/Princess once HM dies I believe, but I suspect Harry will choose not to change what they're known as and follow Edward and Sophie's approach.
Kate and Meghan are princesses, but they're Princess William and Princess Harry, which are lesser titles than Duchess of Cambridge/Sussex.
Camilla is Princess of Wales, but chooses not to use that title for, ahem, obvious reasons.
Camilla will be Queen Camilla and Kate will be Queen Catherine when their husbands become King. Camilla may not use that title - it was announced at the time of their marriage that she would be known as Princess Consort but it is believed Charles wants her to be Queen. Presumably no reason Kate won't be known as Queen.
They will be Queen Consorts (like the Queen Mother was when her husband was King) and not Queen Regnant like the current Queen (i.e., they won't be the monarch). There is no King Consort title, hence Philip being Prince Philip and not King Philip.
I think that's all the points that usually come up on these threads...