The one line answer would be :- because it pays them back in several ways and is a tremendous investment.
Read on for the more detailed answer:
Firstly, Dubai is reliant on tourism far more than Abu Dhabi is.
DXB – Dubai International – has three terminals (1 – which is generally all other long-haul operators), 2 – generally short haul and budget airlines) and 3 – Emirates only and especially adapted for A380.
Be careful now, as it’s often (mistakenly) stated that Dubai is the busiest airport in the world. To measure this is notoriously vague, as sometimes it is passenger numbers, sometimes it is planes landing/takeoff etc etc
Needless to say, Abu Dhabi AUH is a far smaller airport, and even though it has massive expansion plans, remember two things – a) these will take years and b) it’s unlikely they will be busier (whichever way it is measured) than DXB.
Secondly, understand there will be an even busier (eventually) airport to the South of Dubai that’s already open as DWC. Qatar is already flying from there and several other airlines, with more to transfer in time.
The initial aim for DWC is to be the primary connection for EXPO 2020, which will happen very close to the airport in addition to massive development around the area. At present, the airport is virtually in the middle of nowhere but that will change massively and quickly – Fly Dubai is very likely to switch there soon for instance.
Thirdly, understand that although Etihad and Emirates use AUH and DXB as hubs, whilst they are government backed they also have different development plans. Emirates are putting up more planes on more routes, Etihad is investing more into partner airlines and relying on that for growing their capacity.
Dubai is far more of a global hub for business than AUH, with companies based in Dubai needing more travelers for meetings, work, operations and so this in turn creates more demand for flight in/out of DXB than AUH.
So, whilst the government backs both airlines and both airports are developing, the simple fact is Dubai needs more capacity as it has a bigger demand, so in turn the airports are growing accordingly
There is no doubt that Dubai has one of the busiest airports in the world. If you had a chance to land in one of these airports, you would notice long queue in front of the immigration. Even though there are many immigration counters compared to other airports, it’s seen as insufficient. This is due to large number of aircrafts landing at the same time.
Terminal 1 and 3 have limitations for expansion as no space is there for further ground expansion. To accommodate more traffic and people The Sheikh Maktoum Airport is been built. There is enough space for further expansion at the newly constructed airport.
Also the Dubai Government is expecting large amount of visitors for Expo 2020. Thus more investments are being made in aviation sector.