Because it is economic history, it is "economics on hindsight". Meant to shed light on the economics associated with certain historical developments. As opposed to the everyday economics that focuses on prognostics. Prior to it being recognized as a distinct subfield in economics, it was an integral part of historiography. The "new" admits counterfactuals and seeks to make sense of the intertemporal differences in economic measures. Sometimes attempting to facilitate intertemporal comparability of costs and monetary measures through some new proxy or by some new technique. Exemplified by the classics in cliometrics.
I really do not know what is the origin of such differentiation in the way you stated (formally).
However, as long as I remember from Economy classes, the technological paradigm that developed from studies of capabilities, value creation from intangible assets, mainly from Edith Penrose studies (50's - XX century), and Joseph Schumpeter studies (40's-60's of the XX century), have opened a whole new view on Economics. Instead of focusing only on labor and capital as economic factors, technological and resource-based view paradigms tried to understand how same size and same sector firms develop differently, and the technological/innovative/human capabilities aspects behind such different performance. New Economy, therefore, is based on evolutive capabilities of the firms, how they deal with crises and opportunities of the market, and with talent and creativity of their human capital.
In social sciences, each field is filled with ill-defined concepts and terminology that is inconsistently used and leads toward different starting positions. In short, In many publications you will find different definitions, while 'new' is a very tricky word. Think of 'new media' 'or new technologies'.
Economic history is, I think, easy to define, since it refers to a systematic, study of economic phenomena in the past. Since it is a scientific endeavor, it uses several methods to warrant objectivity, validity, and reliability.
New economy, however, reflects different meanings,depending on the angle.
New economy, from an more humanistic approach, refers to the predominance of human values and sustainability instead of mere economic growth. lf the approach is oriented toward production then shift from a manufactory-based toward a service-driven economy is used. And, if te term is used to make distinction in the scientific approach, new economic history refers to more formal or mathematical research methodologies.
The word 'New' is added to mean the difference from the previous one that is old. It is a modern terminology in most of subjects. Sometime 'Neo-' is a prefixed word also been used for the same reason. Actually the related experts are trying to mean something different from the conventional ideas. Both are important to understand different views of a specific subject, even it help most to understand the motive of the new reasoning as well as the motive of this person concern.
Economic history is the study of economies or economic phenomena of the past. Analysis in economic history is undertaken using a combination of historical methods, statistical methods and the application of economic theory to historical situations and institutions. The topic includes financial and business history and overlaps with areas of social history such as demographic and labor history. The quantitative – in this case, econometric – study of economic history is also known as cliometrics.
The new economic history, also known as cliometrics, refers to the systematic use of economic theory and/or econometric techniques to the study of economic history.
Many economists believed that the turning point in economics was the transition between the "zero-sum game" perspective and "free market" perspective that the later was pioneered by Adam Smith. In my opinion, new economics history is focusing on the development of economics since Smith's era whereas economics history is a branch of knowledge in economics that focuses on general development of economics since the earliest time until today.
Just sharing something I read about the fate of economic history and how it is still needed- if not formally in academia, at least as a backdrop in economists' minds/consciousness.