Short Answer:
Rome absorbed and adopted from the Greek almost everything. Including their villages, Greek art, literature, philosophy, and science.
Introduction:
Ancient Greece is the country Greece, located in Europe, during ancient times. It is a civilization that flourished around the Mediterranean Sea from the 3rd millennium to the 1st century BCE (including the first two European civilizations, Minoan and Mycenaean), known as the cradle of Western civilizations.
Ancient Rome is the civilization associated with Rome from the 9th century BCE to the 5th century CE and the Roman empire centered on it.
How did the Greek culture influence the Roman culture?
The Greek influence on Rome started with the settling of Greeks in Southern Italy, such as Naples, (meaning New city). These Naples were founded by groups of Greeks leaving their old culture in Greece, , because of many different reasons, such as trade or overpopulation. This started around the 8th century BC.
This means that from very early on the Italian tribes would come in contact with Greek culture. At this point in time we cannot really speak of Rome yet. The first archeological proof of people living on and around the hills of Rome dates back to around 1000 BC. Mythically Rome was founded by Romulus and Remus in 753 BC (apparently at noon on the 21th of April, precise foes those Romans). Rome only began to be a major force during the Republic and only after the first two Punic wars did the Roman army focus on the Hellenistic kingdoms of Greece.
Although Rome was influenced by the Greek culture already in major ways through the forming of the Roman temples being a good example, , it was after the invasion of Greece the influence became as overpowering as it is seen today. Rome had grown unbelievably in the past centuries, but the growth was mainly militaristic and culturally Rome lagged behind Greece in a number of ways.
For example the "spartan" Romans lived in simple houses, without ornaments and sat down to dinner; while the Greeks had their houses adorned by columns, paintings and statues and ate their dinner reclining. The Romans had yet to establish a flourishing literary tradition or even start thinking about such things as philosophy; while Greece was known for great writers and philosophers like Plato, Sophocles and Euripides.
This different style of life and the cultural wealth of the Greek culture appealed to large numbers of Romans, who started incorporating this in all aspects of their life. It became imperative for members of the upper classes to speak Greek as well as Latin and they usually had parts of their education in places like Athens or Rhodes. This is also the period in Roman history where they started producing their own literature (generally taken to have started around 200 BC with Ennius). Cicero finally translated many Greek philosophical ideas into Latin and invented words like qualitas and moralitas (quality and morality being the English derivatives).
Not everyone was convinced the Greek culture was better than the austere Roman one it was enhancing or replacing. Cato the Elder, for example, was vehemently against all Greek influence and in 161 BC a large number of Greek philosophers and actors were removed from the city, to halt the decay of Roman morals; ultimately this action had little effects