I can speak somewhat regarding the presence of ancient Roman ruins in modern Rome but I know next to nothing about Athens so hopefully someone else can give some insight into that city's development. I will focus on the changes to the cityscape in the first half of the 20th century but others who know more could potentially add information about the role and survival of antique ruins in the medieval and early modern period.

One thing that isn't apparent to most tourists visiting Rome today is that the current form of the city is in large part due to the actions of Benito Mussolini. The boundaries of the city and its neighbourhoods, some of the large streets in the city centre, and the ancient roman ruins the one can see today. Most symbols of fascism have since been removed and streets have been renamed leaving little trace of the fascist role in the city's development to someone walking around.

Mussolini supported the excavation and display of ancient ruins as part of connecting his regime with that of the ancient empire as well as connecting the modern city to its ancient heritage, attempting to craft it into a city worthy of being the capital of an empire rather than a city of parasites as it was considered by some in Northern Italy, including at one point Mussolini himself.

The ruins of the ancient city were not left in isolation during the medieval and early modern periods, some continued to be used as they had, such as some of the imperial bathhouses, some were completely or partially demolished in order to use the stone for other projects, while others took on different roles or were expanded on with later additions. By the time Rome becomes the capital of the fledgling Italian empire the ruins of the ancient city are interwoven with the city's newer layers. The regime's affinity for the ancient ruins meant they were willing to destroy later medieval and early modern developments in order to "liberate" the antique monuments. The later developments were considered as a sort of foul accretion of later city that diminished the value of the original edifices. One such example is the Theatre of Marcellus which had become a mixture of ancient and later buildings surrounded by other buildings of various eras of Rome's history. Much of the surrounding buildings and some of the additions built onto the structure were destroyed with the intent of of opening up the space to allow the ancient building to be appreciated.

The Via dei Fori Imperiali (formerly known as the Via dell'Impero) is one of the main streets in the city centre and well traveled by tourists as it runs from the coliseum and along the various fora. It was built by Mussolini in the 1920's-30's and resulted in two significant changes to the city centre: an entire neighborhood of medieval and renaissance era buildings were demolished to make way for its construction and it allowed for the excavation and opening up of parts of the ancient fora that had previously been buried.

To sum this piece that has run on a bit longer than I intended: much of the antique ruins found in Rome today were unearthed or removed of subsequent additions during the period of fascist rule as part of deliberate efforts to emphasize the imperial history and character of the city, often at the expense of medieval and early modern buildings.

Sources

Arthurs, Joshua. Excavating Modernity: The Roman Past in Fascist Italy. Cornell University Press, 2012.

Kallis, Aristotle. The Third Rome, 1922-1943: The Making of the Fascist Capital. Palgrave MacMillan, 2014.

Painter, Borden. Mussolini's Rome: Rebuilding the Eternal City. Palgrave MacMillan, 2005