The parts that interested me though were those areas held by Assad's forces, those of the opposition, and those which are considered contested. Here is your, "What."
As usual, one of my first thoughts was, "This map is inadequate." So, I went - as I encourage everyone to do as well - to googlemaps. GoogleEarth is better, but run with what you have. This will give us our, "So what?"
There, you have the two maps. See what is going on?
Yep'r, the opposition is smart - they are trying to get control of the cloverleafs on the Damascus version of a beltway.
Control them and the only access to Damascus is from the NW through Highway 1 through the Yafour intersection East to Qudssaya and then in to the city. You also efficiently cut off the airport to the SE.
With limited resources, you can have trouble holding a road - but if you can take, reinforce and hold intersections - then for resupply purposes, the roads are useless.
Sure, we don't have walled cities anymore - but the fundamentals of blockade and siege have held steady for thousands of years.
Now for the "what's next?"
That is up to the players involved. If access to and from Damascus is cut from 3/4 of the cardinal headings of the compass - then it is easier to concentrate on the other 1/4.
Get that - and the city is officially under siege. Modern people and modern cities do not do well under siege. Knowing that - how hard will this fight get? Yea, that hard.
Here is an interesting thing for you to do in your city; what are those Critical Vulnerabilities of your city? What would you need to take and hold? What does an attacker need to take, what can he destroy - and same for a defender.
Do that a bit more and you will get some idea what the Syrians on both sides are up to.
Folks in LA and Atlanta ... good luck with that.