Let's look at the opposition first. The difference between Socs and Greasers is outlined for the reader in the opening chapter by Ponyboy, who is himself a Greaser and thus is able to give an excellent breakdown of what turns one young person into a Soc and another into a Greaser. Note what Ponyboy says:
We're poorer than the Socs and the middle class. I reckon we're wilder, too. Not like the Socs... Greasers are almost like hoods; we steal things and drive old souped-up cars and hold up gas stations and have a gang fight once in a while.
Primarily, therefore, the difference between these two groups is one of class. Ponyboy states that the Greasers are working class, as they are poorer than the Socs. However, in addition, they are much wilder than the Socs, who get into trouble doing frivolous things like "jumping greasers" and wrecking houses when they have a party. This is compared to the Greasers who are more like "hoods" through the illegal activity they engage in. Another difference is the way that these two groups are regarded by society. The press seems to regard Socs as "a public disgrace one day and an asset to society the next," whereas Greasers are largely condemned unrelentingly by society. Lastly, there is a difference in dress. Greasers wear their hair long and also wear leather jackets. These are some of the crucial differences between Socs and Greasers that are introduced in the first chapter.
In regards to your main question, there are of course many similarities between these two groups that are so opposed. Indeed, one of the reasons why this book seems to have been written was to show that differences between Socs and Greasers are artificial, and that actually, underneath those terms, exist young people who are just the same as each other, whether they are Socs or Greasers. One of the ways in which this is conveyed is through the shared interests that both groups have. Both groups like movies and also have a tendency to fight between each other. Both groups too have similar challenges, even though the precise nature of those challenges might be different. Note, for example, what Cherry says to Ponyboy:
I'll bet you think the Socs have it made. The rich kids, the West-side Socs. I'll tell you something, Ponyboy, and it may come as a surprise. We have troubles you've never even heard of. You want to know something? ... Things are rough all over.
This helps Ponyboy to break down the barriers that he feels divide Socs and Greasers and to see them all as young people trying to survive and thrive in difficult situations. Lastly, and linked to the last point, both Socs and Greasers have the capacity to see sunsets and admire and appreciate their beauty. The ability to see sunsets is something of a motif in the story, as it is something that Johnny writes in his final letter as he tells Ponyboy to keep on looking at sunsets. It stands for the capacity to see beauty and to not let the bitterness of life overcome you.