What is the difference between Hell and Lake of Fire? The answers on this question "Heaven/Paradise/Sheol, Hell/Hades/The Lake of Fire" are not clear enough regarding difference between Hell and Lake of Fire.


  And the sea gave up the dead which were in it; and death and
  hell delivered up the dead which were in them: and they were judged
  every man according to their works. And death and hell were cast
  into the lake of fire. This is the second death. And whosoever was
  not found written in the book of life was cast into the lake of fire.
  (Revelation 20:13-15, KJV)


Here is what I understand from reading the above passage. Apostle John was watching the Judgement Day event as if it was happening in front of him. He saw how the dead came to live and how their fate was decided. 

Three places where the dead came from:


The sea
Death
Hell


Three things which were thrown into the Lake of Fire:


Death
Hell
The unsaved people


The word 'Hell' has other translations -


Hell (KJV)
Grave (NLT)
Hades (NIV)
Sheol (Aramaic Bible in Plain English)


The only difference I can find from this verse is that the Lake of Fire is bigger than Hell/Grave/Hades/Sheol because it can hold both death and Hell. :)

How have theologians explained the difference between Hell and Lake of Fire?
In the commentaries and studies that I'm familiar with, this isn't to be taken as two literally different places,this is merely a literary device. (And this coming from a self-described Biblical literalist.)  

For example, from The Pulpit Commentary on the Bible Hub page:


  (2) The sea being a type of the ungodly nations, the sentence may mean
  those spiritually dead, but living on the earth at the time of the
  judgment. The next clause seems to support this view. And death and
  hell delivered up the dead which were in them; death and Hades (see
  Revelation 1:18; Revelation 6:8). As in Revelation 6:8, the two -
  really one - are mentioned separately, the latter being looked upon as
  the guard house of those whom the former has seized. This clause,
  taken in conjunction with the preceding one, may mean - From the
  ungodly nations, those physically living but spiritually dead were
  called up for judgment, and also those who were actually dead, having
  been seized by death and Hades.


Also, from Jamieson-Fausset-Brown Bible Commentary


  
  death and hell—Greek, "Hades." The essential identity of the dying and risen body is hereby shown; for the sea and grave give up their
  dead. The body that sinned or served God shall, in righteous
  retribution, be the body also that shall suffer or be rewarded. The
  "sea" may have a symbolical [Cluver from Augustine], besides the
  literal meaning, as, in Re 8:8; 12:12; 13:1; 18:17, 19; so "death" and
  "hell" are personifications (compare Re 21:1). But the literal sense
  need hardly be departed from: all the different regions wherein the
  bodies and souls of men had been, gave them up.
  


Matthew McGee has a fairly comprehensive teaching on Hell that would seem toagree.


  The English word "Hell" refers to a place of eternal punishment for
  the wicked. Its meaning does not distinguish between the two separate
  places for the wicked to be punished, one temporary for the soul, and
  the other, the Lake of Fire, permanent for the soul and body. Nor does
  its meaning include the place of comfort for saints prior to Christ's
  resurrection. In normal English conversation, "Hell" is used only in
  the negative sense, with no saved people ever going there.