I would argue that the most significant similarity between the Socs and the greasers is that they are both gangs. As such, members of both groups appreciate the value of loyalty, protection, and camaraderie. Members of both gangs are fiercely loyal to each other.

Secondly, it is worth pointing out that both Socs and greasers are around the same age. Many Socs and greasers would probably be surprised if they stopped to think about how much they have in common. The presence of both Socs and greasers at the drive-in theater shows that they have similar interests in terms of popular culture, although it is pointed out that the two groups have differing opinions on the Beatles and Elvis.

Third, both groups are prone to being on the giving and receiving end of violence. Ponyboy recalls an incident some months earlier when Johnny had been severely beaten up and threatened by Socs, leaving him in a state of terror. Another example can be seen right at the beginning of the novel, when the Socs come for Ponyboy as he is walking home from a movie. The greasers are far from above beating up a Soc, and from the greasers’ side, we are told early on that “Dally had jumped people.” This refers to the greasers’ propensity for violence.

The last similarity I will mention is that members of both groups appear to come from unstable home environments. While Ponyboy’s parents are dead, Johnny’s largely ignore him. Poor parenting is also seen on the Soc’s side, and Bob Sheldon’s parents are a great example of this. They have failed to ever discipline their son, give him any boundaries or make him account for his actions.

One similarity between the Socials (the Socs) and the greasers is that they both stick to a singular uniform. Each group has a certain look and style that the members of that respective group follow. When it comes to the greasers, they have long hair. They dress in blue jeans, T-shirts, leather jackets, and either boots or tennis shoes. The Socs sport “semi-Beatle haircuts” and a clean-shaven look. They wear madras, formal shirts, and nice jackets.

Another similarity between the Socs and the greasers is their penchant to define themselves by socioeconomic class. Within the first few pages, Ponyboy points out that the greasers don’t have as much money as the Socs. He says they’re below the middle class. The Socs, too, are defined by their economic situation. Ponyboy describes the Socs as “the West-side rich kids.” He reinforces the relative affluence of the Socs by noting the kind of cars that they drive.

For two more similarities between the Socs and the greasers, think about how both groups exhibit a willingness to fight and engage in deleterious behavior. Furthermore, consider how each group contains members that could be called sensitive or empathetic. The conversation between Randy and Ponyboy at the end of chapter 7 supports the argument that both factions have boys in them who understand that their feud ultimately doesn’t benefit anyone in either party.

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.

The Greasers and Socs are both local gangs consisting of juvenile delinquents who participate in illegal activities and often fight with each other. The Greasers and Socs are both notorious throughout the town and occupy their own territories—where they live and where they hang out. Members of both gangs also enjoy being intoxicated, starting fights, and causing trouble. Although both gangs consist of tough, criminal-minded adolescents, both groups do include sensitive, morally-upright members like Ponyboy and Cherry Valance. Members of both gangs also deal with personal issues and experience their own types of problems.

Despite the many similarities, the Greasers and Socs occupy different social classes and have different reputations among the local authorities. The Greasers reside on the poorer East Side and come from broken homes. They are not as affluent as the Soc members and get a bad reputation among local police. In contrast, the Socs reside in the affluent West Side and have all the opportunities and advantages that the Greasers do not have. The Greasers are also a more close-knit group of friends who are depicted as selfless, compassionate individuals. In contrast, the Socs are portrayed as aloof and callous toward each other.

Superficially, the greasers and the Socs seem completely different; for example, Ponyboy describes greasers as poorer than the Socs and less likely to conform to societal expectations. As the plot of The Outsiders progresses, however, it becomes clear to Ponyboy, and to the reader, that the greasers and the Socs have much more in common than Ponyboy first thought.

The connection between Cherry Valance and Ponyboy that enables them to talk openly about themselves best illustrates the similarities between greasers and Socs. She talks of the tendency of the Socs to put forth an image of aloofness, while Ponyboy talks of the passionate nature of greasers; though aloofness and passion seem to be opposites, they are simply two different ways of experiencing strong emotion. Members of both groups experience strong emotions like love and hate, rage and excitement, but the Socs feel the need to hide it, while the greasers feel the need to act on it. As well, members of both groups feel grief when they lose one of their own, and they feel vindication when they avenge that loss.

Later in the novel, when Cherry acknowledges with Ponyboy that the sunset seen from the West side is just as beautiful as the sunset seen from the East, they share a moment of recognition. This recognition of the similarities between the greasers and the Socs is poignant; though Cherry and Ponyboy understand that they have more in common than meets the eye, that recognition is not enough to prevent more deaths and more conflict between the greasers and the Socs.

Despite their many differences, the Greasers and Socs have several things in common that surprise Ponyboy as he develops into a mature, understanding individual. After Ponyboy has an enlightening conversation with Cherry, he discovers that the Socs have struggles and realizes that everything is not always perfect in their world. Cherry explains to Ponyboy that the Socs have difficulty expressing their true feelings and suffer from constantly having to maintain appearances. Similarly, the Greasers also have difficulties in their lives, which focus more on financial instability and growing up in dysfunctional homes. Both the Greasers and Socs also have family issues. Randy Adderson tells Pony that Bob Sheldon struggled to find someone who would tell him "No," and Johnny's issues involving his parents are illustrated. Both gangs also enjoy thrilling experiences, fighting, and breaking the law. Also, the Greasers and Socs both consist of adolescents who make questionable decisions and are well known throughout their respective communities.

At the beginning of Chapter 3, Pony says that the two types of people are essentially the same, down deep.  But all he really talks about is how they are different.  He and Cherry decide that they are different because the Greasers let their real selves show and the Socs keep things bottled up.  But we're not really told why they're similar.

To me, what makes them similar is that they are both groups of teens who are somehow trying to figure out a way to deal with what society thinks of them and expects of them.  The Socs have to deal with the expectations that come with being rich.  The Greasers have to deal with people thinking they are thugs.  But both groups are really trying to figure out how to be themselves.

I think that's a lot of what this book is about -- teens trying to be themselves while dealing with what society thinks they are and what society tries to push them to be.  Both the Greasers and the Socs have to deal with that.