Whilst it is true that a new runway at Gatwick would be cheaper than a new runway at Heathrow, the "easy bit" ends there.
Every MP around Gatwick has come out against its expansion - this is about 8 MP's but isn't picked up on by the media because none of them are cabinet ministers or have run for London Mayor.
Gatwick will also face legal challenges if it is chosen:
http://www.bbc.co.uk/news/uk-england-37666796
Whilst I have nothing against Gatwick having a second runway I foresee 2 major problems:
Firstly, the majority of airlines at Gatwick are low cost carriers. Any expansion there will lead to an increase in pax charges. The airlines won't absorb that extra cost so it will be passed onto the passengers. The commission predicted that at peak, pax charges could rise from £9 to £30. Any extra passenger charges will more likely affect passengers at Gatwick than those at Heathrow as the later already pay higher charges and aren't generally going for the cheapest option.
The second problem is that even if Gatwick were to build a second runway, Heathrow would still be full. It may just be the airlines trying to persuade government to do what they want, but no airline at Heathrow has said they will move to an expanded Gatwick. You need to remember that all the posturing about extra airport capacity in the South East has never really been about providing extra capacity for the South East, but how to address capacity at Heathrow.
As such there are 3 options going forwards:
If we "need" (desire) a hub airport in the UK then either Heathrow needs to be expanded or a new 4 runway airport needs to be built.
The third option is if we do not "need" (or desire) a hub airport. In this case, the best way forward would be to create multiple equally sized airports around London. As Heathrow blights many people with noise it would be closed and new runways be built at Gatwick, Stansted and Luton.
In my opinion, no option requires that just Gatwick has a new runway. Additionally, I find it unlikely that Stansted and Luton owners would be able to fund such expansion at their sites and therefore would require government funding, at least for surface access. A new airport would require huge amounts of government funding, and considering people are already against spending £50bn on a high speed rail than spans a good part of the country, would anyone really expect people to support spending an equal amount of money on an airport in the South East. Whilst the cost of Heathrow may seem high compared to Gatwick, in my opinion that is a false choice for the reasons stated above. This leaves Heathrow as actually being the cheapest overall option.