Hello, I wrote an answer on a similar theme here. A real problem with this question is that it hinges on the assumption that your average Roman peasant was au fait with classical education in a way that your average early medieval peasant was not, and that that education and knowledge simply disappeared. Indeed Harris and Wright estimate that literacy rates in the Classical world are unlikely to have exceded 20%; likely being around 10% in the Roman Empire (dropping to around 5% in the West), and as low as 5% in Classical-period Greece. By contrast, Sylvia Thrupp estimates that around 50% of the population of Medieval London were English literate, and as high as 40% Latin literate. Indeed, the high number of extant Anglo-Saxon manuscripts suggests that Early Medieval Old English literacy - or at least functional literacy - was relatively high. Part of the question is also predicated on a notion that 'cultures' are mutually exclusive entities which can somehow be deemed materially superior or inferior on an objective scale.

As I discussed in my linked answer, classical education doesn't disappear during the Medieval period. Certainly, texts become more localised, and works were lost, but a great deal is preserved. 'Little Renaissances' like the Carolingian or Alfredian renaissances return a great deal of classical literature to contemporary consciousness, and the foundation of the University of Paris is in part predicated on the vast amount of literature recovered from the library of Toledo following the city's capture in 1085. But Medieval science at literature was perfectly valid in it's own right. Contemporary British/Germanic medicine appears alongside Classical treatments in textbooks such as Bald's Leechbook - much of which has a modern scientific underpinning - and the Medieval period is also known for advances in optics, astronomy, natural sciences and mathematics as well.

One of the most striking Medieval innovations are the vast improvements in engineering and architecture: the Gothic style which evolves in the twelfth century is significantly more sophisticated and more effective than its Romanesque predecessor. I wrote a post in more detail about the facets of Gothic here, but the TL:DR is that the architectural advances of Gothic allow for thinner, stronger and taller walls, stronger vaulting, bigger windows and stronger pillars, which leads to buildings which are much bigger, airier, brighter, stable and ornate than Romanesque equivalents.