There's really a few questions being asked and it would be better to separate them.
Why didn't Europeans in the Middle Ages appreciate "antiquities?"
First, because archeology hadn't been invented yet and two because it was the Dark Ages. Now, the Coliseum was scavenged because so much knowledge was lost after the collapse of the Roman Empire, the people really needed the material, but there other things at play as well. During the early Medieval Period, or "Dark Ages," through the Renaissance, art was regulated by the Catholic Church. A good amount of very beautiful art was created, but its intentions were to help its audience spiritually. All writing, music, painting, drama, sculpture and architecture created during the period reflect these notions. What was art and was not art was something decided by religious clergy. They could be particularly hard on dramatic performances, multitonal music or any developments in art. At the time, they absolutely did not consider pagan philosophy or art to be appropriate topics.
In the Renaissance (14-17th century), the attitude lightened a great deal. Pagan philosophy and ideas were taught in universities. Nonreligious topics were allowed in art again for the first time. The movement originated in Italy and there was also interest in studying the old remains of classical Roman buildings.
Filippo Brunelleschi was foremost in studying the remains of ancient classical buildings, and with rediscovered knowledge from the 1st-century writer Vitruvius and the flourishing discipline of mathematics, formulated the Renaissance style which emulated and improved on classical forms. Brunelleschi's major feat of engineering was the building of the dome of Florence Cathedral.
Why could antiquities be purchases so cheaply, so recently?
Short Answer: Because they were stolen (or collected) from nations that were European colonial possessions. Also there was a misunderstanding of the best ways to perform archeology in general.
Long Answer: Europe emerged rather quickly from the Renaissance with several nations competing for global empires. These developments influenced European art and philosophy. They were very curious about other cultures and thus the European aristocracy and nobility tried to add the best artwork from around the world to their private collections. In some instances, such as Napoleans conquest of North Africa and Egypt, the art and antiquities were outright stolen through war. In other instances, they were likely gifts, although as you will see below, today it is not considered possible to actually gift a nation's most prized artwork.
So, the influx of such large amounts of foreign antiquities made the price lower than today. Also, now that these colonies are independent nations, they have rightly requested back their artwork, so it is no longer on the market. This increases prices a great deal.
As to the second point, I believe Dolda2000 answer is satisfactory. It simply isn't allowable to sell artifacts piecemeal as was once done, since it ruins the context for the artifact. Research is much more profitable if the researcher has all the artifacts and if they haven't been tampered with. Therefore, the market for antiquities is again restricted.
Why are they consider "priceless" now?
The modern appreciation and value placed on historical and artistic artifacts is derived from the creation of the modern nation state. In order for a people to understand who they are, they must have a preserved shared past. Examples from the late Middle Ages and early Modern Era in France and Germany will be instructive. In modern times, many (if not all) artifacts are now literally considered priceless (even if sometimes they can be purchased at a price). This is because they belong to "the people" and "posterity" and this idea comes from the French Revolution. The price for private collectors is driven up since most artifacts will be in public museums.
The French Revolutionaries converted Louis XVI palace into the famous Louve museum. While museums did exist before this, they were somewhat rare. This action was a statement that the rich and powerful do not have the right to keep art for their personal enjoyment. It belongs to us all. This idea drives laws and restrictions on purchases on art and antiquities.
Source: Sherman, Daniel J. and Rogoff, Irit, ed. Museum Culture Histories Discourses Spectacles