I'll add a slightly different slant and suggest that the central conflict in The Great Gatsby is illusion vs. reality.
The novel is first and foremost about Gatsby. It's named for him and it is about him. When Nick concludes the novel he does so by referring to the subject of recapturing the past, which is what Gatsby tries to do the entire narrative. That is what the work is about. I suggest that everything else is secondary.
The book is above all else a love story. Gatsby's love for Daisy is what makes it worth reading, the same as Shakespeare's Romeo and Juliet is primarily a love story and that's why it's still popular. The general population doesn't watch Romeo and Juliet to see what it has to reveal about feuding. Numerous other issues exist in that tragedy, too, but it is above all else a love story. Gatsby's story is the same.
Gatsby's view of his relationship, however, is an illusion. His love for Daisy is everything he thinks it is, but Daisy's love for him is not. He spends five years of his life chasing an illusion, and he refuses to give up on it even when all rational thought and all evidence suggests that his illusion is false: of course he does, he's in love. Society cannot stop Gatsby, neither can money. Gatsby has money. If that's all there was to it then Daisy would have gone with him. Again, Gatsby has money--Daisy wouldn't have to give up a thing as far as money is concerned. Which man has the nicer shirts? Gatsby, of course.
Daisy does not go with Gatsby because the illusion in Gatsby's mind is false. She refuses to say that she never loved Tom. She refuses to say that she's been pining for Gatsby all these years. He asks too much, she tells him.
The only thing that can stop Gatsby is reality. Reality trumps illusion in The Great Gatsby.
There are many conflicts in Fitzgerald's work. I would say that the primary conflict is between individual and society, how Gatsby seeks to define himself and immerse himself in a social setting. Along these lines, I think Nick faces a similar conflict in which he has to choose the type of person he will become from his experiences with Gatsby, Jordan, and the Buchannans. The conflict present that seems to impact Gatsby, and Nick, is to what extent they will go in order to become part of the alluring world of social mobility. Their conflicts manifests itself in many ways, but specifically resides in the idea that there are certain aspects of the social order that might prove morally challenging or difficult to ethically accept. This conflict of someone on the periphery seeking acceptance by the central majority affects Nick and Gatsby, with the latter having to deal with the painful aspects of it and the former learning from it.
I think the over-reaching conflict that goes from character to humanity is the conflict between character and materialism.
Gatsby would have had Daisy from day one if money or material objects and status didn't matter, but to many of us they do. We know we don't want things to be that way but they are. If Daisy as a teen didn't see Gatsby for his poverty but for the person he was, he might never gone as far off the deep end seeking her approval.
I think the conflict of the story reappears in many ways. Myrtle likes Tom for what he can buy her and therefore embarks on an affair.
People enjoy Gatsby's parties because they are free - he provides these material items people wouldn't otherwise always get themselves.
Money makes people do things they wouldn't otherwise do.
To me, the main conflict in this book is between Gatsby and society.
For all the time that we see him, Gatsby has one goal -- to get Daisy. This is really all his life is about. She is his dream.
But society has sort of put Daisy out of his reach. Because she was born rich and he was not, she did not marry him. Because he was not born rich, he had to resort to criminal behavior to get rich enough, fast enough, to be able to impress her.
But this is still a problem because society does not approve of what he has done to get rich. So Gatsby has been fighting against society all his life.