The Oxford English Dictionary recognizes no difference between hug and embrace in their primary meanings, and neither does not the source you are apparently quoting.
I think the answer that says The main difference is the level of affection shown in each is being too strict regarding hug and embrace. Now, hug may have a wider application, as in the bear hugged the man before it killed him (!) (Note: we would not use embraced as a synonym in that sentence); nevertheless, regarding human hugs and embraces, I'll say there may be some trend or thought in some people's minds, but this is not set in stone.
The Oxford English Dictionary defines hug
I. 1. a. transitive To clasp or squeeze tightly in the arms: usually with affection = embrace
It also notes the figurative usage:
I. 1. d. figurative To cherish or cling to (an opinion, belief, etc.) with fervour or fondness.
however, I agree that nowadays I have probably only heard embrace used in this way. But the OED does not mark this as obsolete or archaic or anything.
Then there is the usage meaning 'to cling to', used for ships and other things such as pathways.
4 transitive (orig. Nautical) To keep as close as possible to (the shore, etc.); to ‘cling to’.
Embrace does not have this last meaning, as far as I am aware. Interestingly the etymology that the OED gives for hug is..."unknown".
As for embrace
, the OED provides
1 a. transitive To clasp in the arms, usually as a sign of fondness or friendship.
There are many figurative uses, including
2 h. To adopt (a doctrine, opinions, religion, etc.); often with the notion ‘to accept joyfully’. Also, to attach oneself to (a party, cause, etc.)
which is similar to one of the figurative uses of hug.
But as for the literal meaning in terms of to clasp in the arms, there is no description in the OED that says that one is more affectionate than the other.
Indeed, the OED flatly states hug = embrace.
So feel free to embrace your friend or relative and hug your lover.