Seems likely that they will parley with each other first but fail to come to an agreement about the throne. Aegon will suggest that his claim is superior (believing himself to be real) and probably will already be on the Iron Throne. Whatever he offers he will not be good enough for a queen that was so powerful in Essos and who is bent on claiming Westeros as her own domain. Daenerys's allies will likely urge a militant approach to dealing with those who doubt her legitimacy or overriding right to the queenship.
The fact that both of them will probably be married will present a huge problem also that will be one of the primary catalysts to conflict. Another would be if Daenerys suspects or somehow knows that Aegon is a fraud (whether or not she specifically knows of the Blackfyre stuff). It would be easy to be suspicious of Aegon's claim to be a real Targaryen but it would be extremely difficult to actually provide evidence that he is false, although apparently somehow that will happen, at least in a way that is convincing to Daenerys herself (slayer of lies, etc.).
As many others have said, Aegon will probably either marry or make a marriage pact well before Daenerys enters the picture. I suspect it will be before he gains the throne and that it will be done in part to secure valuable military support. A new DWD could be avoided if the following conditions were met:
1. Aegon remains unmarried long enough for Daenerys to arrive and has achieved a significant level of success without having to rely on a marriage alliance. (No longer nearly as great a strategy from the Golden Company's point of view; their planning now no longer focuses on Daenerys's help and their Maester is talking about Aegon marrying someone in Westeros).
2. Daenerys remains unmarried or willing to engage in a politically motivated divorce. There are several things which would likely derail this. There is the question of who will follow Hizdahr; perhaps she will need the help of the Greyjoys' fleet. Then there is the problem of whether she will be convinced by the propaganda which alleges that Aegon is who he claims to be.
3. The two would have to settle on how to co-rule. Both of them essentially want to be the predominate element in ruling Westeros. Neither would accept subordination to the other.
Probably none of these three conditions will be met (though point three would be reasonably possible if points one and two were achieved) which will make the initial negotiations quickly turn antagonistic. This in turn will cause each side to mobilize and prepare for a conflict which will become inevitable.
Despite this being a massive, large scale conflict, I think that there will be yet more facets of the war for control of Westeros even after the collapse of Aegon. Perhaps from Stannis or other sources.
Edited by The Sunset King, 09 August 2012 - 06:48 PM.