Small Open Economy (SOE) Vs. Emerging Market Economy (EME)

These are not synonyms and actually the terms have different meanings (with small overlap), but it is true that many emerging market economies in many settings just happen to be small open economies.

Small Open Economy

First, open economy is any economy that is open to international trade (see the usage through various textbooks such as Blanchard et al Macroeconomics).

Second small here means economically small, that is the economy is a small economy if it cannot affect world prices (in the relevant problem). Small country here is an opposite of 'large' country which is a country that can affect international prices (note this has nothing to do with geographical size of a country). This is pretty standard use of small country in international trade (see Krugman et al International Economics pp 197 for examples of this terminology in use).

Emerging Market Economies

Emerging market economies are (see the Economist explainer):

broadly speaking, an economy that is not too rich, not too poor and not too closed to foreign capital.

This term was originally coined by Antoine van Agtmael in 1981, it was supposed to serve as better term for economies that are in between being under-developed and being developed economies.

---

Now note there is an overlap between these terms as many emerging markets happen to be small and open economies, but there might be exceptions. For example, China is likely sufficiently large producer of many goods and services that it could affect world prices and thus be a large country in many industries yet it is classified as emerging market by IMF. However, the both definition partially overlap in that emerging market economy definition also requires certain level of oppennses.