How does the introduction of victory cities in Axis & Allies revised change gameplay and strategy?
With the introduction of "victory cities," the revised edition tilts the game towards Asia and the Pacific.

Seven new cities have been added to the original roster of five (Tokyo, Washington, London, Berlin and Moscow). The new ones are Paris, Rome, Leningrad, Los Angeles, Calcutta, Manila and Shanghai.

Of these, Paris, Rome, and Leningrad had natural importance in the early editions, en route to Berlin and Moscow, respectively. The Allies actually captured Rome and Paris on their way to Berlin, and arguably, the Germans should have secured Leningrad before attempting Moscow. In a sense, their importance had not changed, only their "scoring."

On the other hand, Los Angeles, Manila, Shanghai and Calcutta have now taken the importance of the other three cities. They are worth fighting for in their own right, rather than just for a few IPCs. Put another way, there are 166 IPCs on the new board and 12 victory cities, meaning that pro rata, each victory city is equivalent to about 14 IPCs. In the old edition, only Los Angeles had anywhere close to that importance. Now it makes sense for both the U.S. and Japan to fight a war in the Pacific instead of heading their forces toward Europe.

The other important change is victory conditions. In the old editions, the Allies were forced to aim for "total victory"  while the Axis were encouraged to go for a major victory (all except the U.S.). (Both of these would be after the implied "mopping up"). 
There are now minor and moderate victory conditions. The Axis can win a minor victory early by capturing, say, Leningrad and Calcutta, which is more than adequate compensation for the loss of the economic victory option. If the Germans had captured Leningrad and Japan had gone for India instead of Pearl Harbor, the Axis might have gotten a "negotiated" peace.