Important attributes that the Romans felt separated them from other "barbarian" peoples included hygiene, language, and the building of cities. Roman grooming habits were different than those of peoples who lived to north of them. That is not to say that other peoples did not clean themselves, but the did have a unique odor that the Romans disliked. This is largely thought to have originated from the use of different types of oil in cleaning and grooming. Romans, and many Mediterranean peoples commonly used olive oil to clean themselves, and would lend them a certain smell, which they grew accustomed to. Celtic and Germanic peoples living in northern Europe more commonly used animal oils for the same purposes, and made them smell differently. So when the Romans encountered these people, they thought them smelly, unclean, and barbaric. Also, around the time of the Punic Wars, the Romans began shaving their beards. It was quickly adopted after the hero Scipio Africanus began going about clean shaven and remained in fashion for centuries. Many other peoples did not follow this custom. Additionally, Greek was generally considered the language of educated people in ancient Rome, and Latin eventually grew in prestige as time went on. If you did not speak Greek, or at least Latin, the Romans were prone to look down on you as barbarians.
Finally, the lifestyle of the city set Romans apart from many other peoples they encountered as they expanded their empire. Roman cities were unlike those of other peoples. They did not usually grow up organically, but were deliberately built on a grid pattern. They would be supplied with water by Roman aqueducts and contain important buildings in the Roman lifestyle, like baths and Roman temples. The Romans prided themselves on their advanced architecture, not without good reason. These things did largely set them apart from other peoples. In many ways, they shared these attributes with the Greeks, but the Romans typically did not label the Greeks as barbarians. As the empire aged and more people who were not of Roman descent but who lived in the empire adopted these aspects of Roman culture, the idea of who was a Roman changed. Eventually, Roman citizenship was expanded to include most freeborn men who were born in the empire.