Is there any real advantage of flying into London in either one?
If given the choice, I will always use Gatwick. Mostly because it is a smaller airport and easier to navigate. The Gatwick Express train takes 30 minutes to get to Victoria Station whereas the Heathrow Express train takes 19 minutes (from terminal 4)to get to Paddington Station. But Victoria Station is a little bit more centrally located then Paddington Station. Also if your final destination is outside of London, Gatwick may be a better choice as its connected to an inter-city rail line, whereas Heathrow is not.
There are advantages to each airport, so you might want to figure out what is best for you. You can take the tube into the city from Heathrow (40 minutes) or an express (15 min to Paddington). The tube is obviously cheaper, but Heathrow is the 3rd busiest airport in the world. I like Gatwick as it is much easier to manage and navigate, but you must take a train into the city (40 min) and the cost ranges from £8.90 (Southern one way; return is about 12) to £15 on the express which is only 3 min. faster and both come into Victoria station. I think Heathrow is slower to get through customs than Gatwick, but that might differ day to day. Also, prices on tickets differ from airline to airline and I usual pick whichever ticket is cheapest but personally prefer Gatwick.
Assuming you're not making a connection and London is your destination, probably not. If you have a choice of ticket prices, go with the cheapest. If the price differences are insignificant, go with the best schedule.
Heathrow is a bit closer to town, so transport into the city might cost a bit less. Plus, there's a Tube line. Otherwise, the options at either are pretty much the same: an express train, local trains, buses, hired cars (a driver meets you and takes you where you're going) and cabs.
Passport control can get backed up at any busy airport, especially if several wide-bodies arrive together, so that may be a matter of luck.
The flipside is departing. Heathrow has earned a pretty bad reputation this summer for losing luggage, hours-long queues, missed flights, and general chaos. Gatwick isn't heaven, but the press it's getting isn't as bad.
The advantage is what is better for you. If I had two itineraries that were about the same elapsed time and cost I would take Gatwick but if Gatwick were a longer itinerary and a bit more expensive, Heathrow. Heathrow is okay to fly into (although you have to walk some LONG distances), the problem is flying out of it.
I did the exact same route using each airport, and LGW was $300 less than LHR.