I think they are synonyms. "Hug" is the ordinary word used in everyday speech. "Embrace" is more formal and literary. Of course, "embrace" has a much wide range of meanings. Double-checking with my favorite dictionary, ahdictionary.org, the relevant definitions seem so be "hug: To clasp or hold closely, especially in the arms, as in affection; embrace," and "embrace: To clasp or hold close with the arms, usually as an expression of affection." That sounds almost identical to me. I will sometimes ask my wife to "give me a hug." I can't imagine asking her to "embrace me," despite the lovely 1930 popular song, "Embraceable You." This song, by George and Ira Gershwin, begins: "Embrace me, my sweet embraceable you Embrace me, you irreplaceable you," but I've always felt part of the charm was the _unusual_ word "embraceable," and the amusing discordance of using slightly formal language to describe intimate feelings.

Hug and embrace are very similar. However, hug refers to the action of holding someone tightly while embrace is broader and means to hold someone closely (more affectionate).

Hug is more casual while embrace suggests a more intimate relationship. Embrace can also mean more than just a hug. I would say friends hug and lovers embrace.