You could go with a Manifest Destiny style approach. Convince the people in the more advanced society that they will be bringing civilization to the poor, uneducated, savages, and they'll jump at the chance to conquer them and keep their land.
You don't need to go to fiction for this, either. Just look at the history of "The Irish Question". 700 years after the English conquered Ireland, the British public were still being fed much bullshit about how the Irish weren't fit to govern themselves, were uncivilized brutes, and in need of the loving hand of Mother England to gently teach them. Never mind the numerous armed rebellions and the fact that a majority of the Irish didn't want them there. Please note that I am not talking about The Troubles or Northern Ireland (I don't want to start a politically charged firestorm here, after all). This is all stuff over a century old, i.e. before the Anglo-Irish Treaty and the creation of Northern Ireland.
For a more recent example, what about the creation of Panama? America wanted the Panama Canal built, so they gallantly assisted the locals in their drive for independence! Go America! Ignore the fact that Panama was a part of Colombia!
There's also the Mexican-American War. In a nutshell, Mexico owned land that the US wanted. After the bitter fight over Texas' independence from Mexico, most Mexicans were disinclined to negotiate for a straight up purchase of the land. Both sides got themselves worked up to the point that fighting broke out. After that, the honor of the nation (on both sides) required war. End result? The Treaty of Guadalupe Hidalgo and the Mexican Cession. The USA gets California; Utah; Nevada; parts of Colorado, Wyoming, Arizona, and New Mexico; and the Rio Grande set as the boundary between Texas and Mexico.
TLDR version
Nations having the same (or similar) cultural values is no bar to war. If nation one is more powerful and wants something nation two has, a casus belli can always be found or manufactured. If the more powerful nation can convince its'its people that the war against—or conquest of—nation two is "for their own good", or that nation one deserves the land more than those savages do, it makes a war so much easier to start.