Arguably, the difference is based on whether you would have supported establishing Israel because it was a mitzvah or whether you would have gone along with it since it benefitted the Jewish people. If you are under 80 years old (i.e. weren’t involved in politics in the 1940s), this discussion is moot.
In Israel, at one time you could say that a “Hareidi” was someone who held that the government of the State of Israel had no more halachic status than any non-Jewish government, and it was permitted to ignore any of its laws that were contrary to halacha. Modern Orthodox held the State had a special status under halacha and its laws were binding. After Gush Katif, it is arguable that anyone “modern” still holds that way.
The lines are totally vague and are based on distinctions that are largely historical, especially in America. Especially in America, many people with beard and pe’os, who dress “frum” hold college degrees and work as professionals. In both America and Israel, but especially in Israel, there are many people with a kippah serugah who say hallel on Israeli independence day, who also are strict in all mitsvos and learn Torah full time.