What's the difference between an atomic bomb and a nuclear bomb?

They refer to the same thing.

Basily an atomic bomb uses a fission reaction created by the decay of a large number of atoms of a heavy element. The small amount of mass turned into energy causes a massive explosion.

A hydrogen bomb or thermonuclear bomb uses a fission reaction to initiate a much more energetic fusion reaction in which atoms of a hydrogen isotope fuse to become helium and the small (but far larger amount than in fission) of mass turned into energy causes an even more massive explosion.

Both are nuclear bombs or weapons, as are neutron bombs and other variants (for examble trinary bombs have a three stage nuclear reaction) because they use a nuclear reaction to create an explosive effect.

There is no differences. This is extremely stupid invention. Einstein was working on it.

actually they are both bomb with great effects... they are famous of different times... atomic bomb used last world war 2, nuclear bomb is now on testing for future wars...

They are the same.

Only Farron has it right, they are NOT the same.
There are two type of processes that can release the energy inside an atom: fusion and fission.
Fat Man and Little Boy were the two atomic bombs used at the end of WWII.  Modern nuclear reactors in power plants and some vessels are fission systems.  The first sustained fission reaction was created/performed by Enrico Fermi on a squash court. Fermi used "control rods" to keep the released energy levels at a usable state[which is what modern reactors do as well], the atomic bombs released all the energy at once.
Today's modern "nuclear" bombs generate their destructive power through the fusion process (again like Farron said they generate energy to create fusion by first creating a fission reaction).  This is what the Sun is doing to convert approx 600 million tons of hydrogen into helium every second--except without any fission first.   The fusion process is what makes all stars shine.
And, Einstein never worked on the bomb.  He even signed a letter written by Fermi and others and addressed to FDR saying how unimagineable the destructive power of an atomic device would be. The letter said that type of weapon was wrong and immoral.  But, this was the letter, since it said the nazis were working towards making their own, that caused the Manhattan Project to be commissioned on Dec. 6, 1941--yes, BEFORE Pearl Harbor.
And, just how does one calculate how much energy will be released from a given amount of matter? Simple, the conversion rate is c, as in the famous equation E=mc^2.

There is no difference. They both use Uranium or Plutonium fissile fuels to create nuclear reaction and the byproducts of the reaction help to maintain the reaction, in turn creating a chain-reaction.