I have a daughter currently at Cargilfield. My son was also a Cargundian.
I have found no difference between them.
If you are considering Cargilfield seriously then it is worth bearing in mind that their primary objective is to prepare boys and girls for the top Public schools in the UK. It doesn't matter what sex your child is, they will prepare them for the best school for them as an individual. The major difference between Cargilfield and the other 2 schools mentioned is that it gives you as a parent and the child options. As your child grows and develops it allows you to see where their strengths and weaknesses are and when the time comes to move on, you are confident you are choosing a school that best suits your child at that crucial time in their life, that school might be Lorretto or Fettes, it might be Eton (for a boy) or Wickham Abbey (for a girl) the point is you have a choice. If they start in an all-through school they will not be prepared for Common Entrance or a Scholarship to another school as it is not in their interest to do so. In a 3-18 school they are also in the same small cohort of pupils without the opportunity to in effect re-invent themselves when they start at secondary school too which in my opinion is a real shame and can feel very claustrophobic.
The draw back of Cargilfield is if you decide your child is not heading to a major public school or independent. The day schools in Edinburgh all have their major intake at 12, as many kids come from state school at that time so you need to withdraw your child from Cargilfield a year before the rest of their class mates or take a chance that a place will become available by going on the day schools waiting list (not ideal as your child then doesn't have the security of knowing their destination school) if you leave them at Cargilfield until they are 13. The Curriculum they follow is also different to Edinburgh day schools i.e Cargilfield - Common Entrance and Scholarships, Day Schools - Curriculum for Excellence.
I am, however, an enormous fan of Cargilfield. I cannot stress enough how fantastic a school it is. From Form 4 onwards the children have separate specialist teachers for each subject (just like high school), the most able are accelerated and stretched whilst the children needing more support have a very nurturing environment. This is a competitive school environment though, competition is hugely encouraged and success is celebrated in all areas, sport, drama, academics, music. It is full on, certainly in the upper school but very enriching