One notable difference, too is that "modern" astrology, as we know it, really sort of stemmed from the 1700s and the Age of Enlightenment - as it was. Everything at that time took a sort of swing toward a scientific, "rational", practical and mechanistic perspective. This meant that if it wasn't scientifically explanable or rational then it wasn't valid. if it could not be held, dropped, proved, demonstrated and touched then, it wasn't real. I suppose Voltaire tried to bridge this developing rift between the natural and deific and the scientific and rational but it was a heavy tide to contend against.
So, as new things were discovered, astrologers felt that in order to protect the integrity of astrology, the new scientific ideas and discoveries had to somehow be grafted on to the foundations of astrology (i.e. Uranus, a new planet now needs a sign to rule- shouldn't ALL planets have signs to rule?).
Of course the more the solar system grew the more astrology made room for it all and slowly, the wheel became more disfigured as we go along.
So, now we have the 'modern' approach which is simply adding things into a chart and guessing how to read them. There is no particular form or method to do this. It is just done. Some people pick 'these' things and others pick 'those' things. The truly courageous just add everything to a chart and then, in retrospect, make it work with known outcomes.
Modern astrology has value but no practice or discipline and is relatively new in light of the history of astrology as well. Even Sun-sign astrology is not astrology but simply an invention popularized during the WWII time period but, other than that, has little historical bearing.
Traditional astrology with all it varying methods is, now, becoming used like modern astrology is. You see people who say they are traditional astrologers taking Hellenistic methods and mixing them with modern house systems adding Arabic parts and then, say, quoting Lilly. It's just a hodge-podge after so much of this.
So, what is the difference? Well, true traditional astrology adheres to a set of rules in a specific context. Something modern people in general rue. It also has a philosophy or structure like seven planets rule the twelve signs and do so for the following reasons...etc. etc.
On the other hand see Uranian astrology. Very modern. Now that will freak out even the most open-minded modern astrologer. Yet, it works! But, only when traditionally used in its own context and set of rules that it developed under. You can't just take the trans-neptunians and use them the same way out of the Uranian system and get the fine results or anything truly useful or predictive.
I guess, in short, what I see, is that there is a system and a method or discipline that has to give form to whatever is being used rather than discarded in favor of a free-for-all. Astrology seems to break down in that sort of formless void.