This is what I know of the actual history.
At right around the same time period as the formation of the Golden Dawn, there was an Antiquarian group that was mainly interested in the study of occult books, mss, etc. They were not a practicing group. More of a interest group. From within that group a number of members decide to form a ceremonial order. i.e. a practicing group based on what they had been studying in theory.
The Golden Dawn and Aurum Solis have never been connected in any formal way. The founders were different people although there may have been some overlapping members at times.
Almost all Magickal orders share some common thread in the overall focus of their practices. I think both could be described as somewhat Hermetic in practice. Here is where I see some differences forming. If we move chronologically from the beginning of time forward in the chain of ideas of the two orders share many of the same neo-platonic ideas. Not the actual orders themselves but what there influences are in chronological order.
A bit of a split of focus occurs in the ideas they focus on when we hit the 15th-17th century. Here the Golden Dawn takes on Rosicrucian ideas with both Christian and Judaic mysticism. Therefore we see the influence of the German mystics of the time, and it continues from there under the influence of the Kabalah, freemasonry,etc. So at it's heart the Golden Dawn is carrying on the Rosicrucian tradition or current that was first made public by the Manifestos in the early 17th century.
The AS leans more towards the Italian Renaissance philosophers. The Ogdoadic tradition springs from a group that was formed by one of the Medici's and included some of the finest Italian thinkers of the time. I think it was called The Order of the Helmet. But don't quote me.
So the A.S. looks much more into ideas put forth by those who they consider part of the "Ogdoadic Tradition", Pico de Mirandola, Marsilio Ficino, Nicholas of Cusa, Giordano Bruno.
I would really encourage you to listen to the podcast link I posted as one of the heads of Astrum Sophia a chartered group of the Aurum Solis explains the question of the differences. He lays it out quite well and this would give you it right from the horse's mouth so to say.
I can only speak from my own observations, I have never been a member of either group so I don't want to lay out too many ideas on something I may know nothing about. I have worked with an order before and I found that the overall quality and focus of a group is largely dependent on it's members. Neither of these groups "forbids" any studies that the other shares as far as I know so they aren't actively restricting ideas they don't focus on.
One interesting thing is that the heads of the Aurum Solis and their chartered offshoots have repeatedly stated in interviews that all of the documents and teachings of the Order are public. So that by joining you would not be exposed to materials that are not available to anyone. Joining provides mentoring and group practice but it does not open any secret doors of knowledge. I don't know if the Golden Dawn has materials that are considered "Order only" documents and it might vary from GD to GD. The O.T.O. definitely has things that are meant to be obtained only by joining.
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"For many years I have been a Lapsed Idiot. With faith and penance, I hope one day to be a devout Imbecile again." - chaoscrowley
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