So what’s the difference between a sobriquet and an epithet? Would the earlier example of Richard the Lionheart be a better example of a sobriquet or an epithet? Wiki suggests that Richard the Lionheart is actually an epithet, while something like, say, The King (for Elvis Presley) would be a sobriquet.
And there there’s: username. Sometimes called nom d’electron. 
Not sure if the OP wanted grammatical differences, but even in the posts above, the “nom” terms are called “loan words,” in that they are borrowed from another language and not fully adopted into English. So, when used, they are usually in italics and do not take any additions like plural -s. The rest of the words are regular English words or compound nouns and conjugate accordingly. Many words start out as loan words and over time become regular English, e.g. tsunamis.
So technically, they would not be interchangable as loan words generally require the italics while regular English would not.
If we’re being absolutly anal, an epithet has to be an adjective descriptive of the person concerned; Charles the Fat; Gregory the Great; Fat Tony. (Whether “Lionheart” is an adjective I leave it to the grammarians to argue about.) It’s probably safe to say that all epithets are also sobriquets, but the converse is not necessarily true, since any nickname, whether or not adjectival or descriptive, can be a soubriquet.
But if we’re not being quite so anal, there is a well-established extended sense of epithet covering any signficant appellation, not necessarily of a person. And that would cover all sobriquets.
Thanks USD for those definitions. Thank you all. Very helpful.
davidmich
Its not so simple when dealing with Middle Eastern naming conventions. Some are regional. Some are tribal. For instance in Iraq you can use any one of multiple combinations of your name, your family members names, your tribes name and your profession. All are perfectly valid. Saddam Hussein Abd al-Majid al-Tikriti could have been known as Saddam al-Tikriti and it wouldn’t be a pseudonym.