Good question!
Sorry-long post alert...
Atomic-nuclear-same same, more or less. 'Atomic' bomb has sort of fallen out of favour, as 'nuclear' sounds 'sexier', though there is, in fact, a distinction. To start off, there are 2 types of nuclear reactions to consider. Fission, atoms spitting apart to release energy in a really big bang, and 'fusion', fusing atoms together to release energy in a smegging gigantic kerblammo , the energy released as a by product of 'reorganising' atoms of one material into another. Note, this is a pretty simplified explanation, particularly with regard to variable yields and such.
A straight 'Atomic' bomb is a pure nuclear fission device that relies on run-away fission, that is, 'splitting' Uranium-233/ 235 or Plutonium-239 atoms apart to release energy in the form of heat and high energy nuclear particles; literally, lots of atom blowing up like wee grenades, cooking off the other wee grenades around it . These 'fissile' fuels are metals that are pretty unstable at the atomic level, they 'decay' as the atoms fall apart, emitting neutrons, and this kinetic activity(particles smashing together) generates heat. These atoms can be easily broken apart by a relatively slow moving neutron, releasing more energy and a shower of neutrons which break up more of the fissile fuel atoms, creating and sustaining a 'chain reaction' when you assemble a sufficient mass of material, a 'critical mass'. A critical mass can be arraigned so that the reaction is 'moderated' by controlling the rate of neutron emission, a nuclear reactor. The critical mass can also be smashed together rapidly under huge pressure with conventional explosives, resulting in a huge number of fissions, all at once- the atomic bomb. After that, the bomb effects are simply a really frikken huge deflagration, overpressure, heat into millions of degrees at close range, with radiological effects from the storm of high energy particles-neutrons, protons; and various 'rays'(more high energy particles of various half lives and energy levels) X-rays, Gamma radiation, Beta radiation, and secondary/tertiary radiations from the atomic structure of other matter being rearranged-turned into isotopes-through high energy particle bombardment. Radiation and high energy particles 'kill' by literally kinetic action at the atomic level, destroying tissue and chromosomes.
An Atomic bomb can be the simple Uranium-235 'gun' type, famously, the Y-1852 'Little Boy', the weapon that severely damaged Hiroshima, and formed the early US nuclear stockpile. This type of 'bomb core' is really simple to achieve and manufacture, but is inefficient in its use of nuclear material, is limited in yield, and can be susceptible to un-commanded detonation in an accident. The Y-1561 plutonium implosion core design,the 'Fat Man' that was dropped on Nagasaki, was a much more difficult weaponeering problem, but used fissile fuel much more efficiently, and can produce much higher nuclear yields-power, as well as making a better weapon in terms of compactness, ease of handling and safety. The plutonium implosion core design formed the basis of the 'productionized' US nuclear stockpile, and was subsequently copied by the Soviet Union through espionage, and then supplied to China. The Pu-239 implosion core is the 'standard' nuclear design among the worlds nuclear powers, and is the trigger for a Hydrogen, or Thermonuclear bomb (H-bomb).
A 'Nuclear' bomb is a generally thought of as plutonium implosion bomb that has its output (yield) 'boosted' with low order 'fusion' reactions by injecting tritium gas into the bomb core on detonation. This allows very high yields with out using a very large amount of plutonium in the core.
Typical yields for an Atomic bomb are measured in equivalents of a thousand tons of TNT, typically sub-kiloton to 100 kilotons, though high yield pure fission designs use fissile fuels very inefficiently.
A modern nuclear bomb core tends to be a minimum critical mass, fusion boosted to increase the yield of the weapon. The most modern weapons are 'dial a yield' where the explosive power of the weapon can be selected by modulating the amount of tritium and the timing of the injection, changing a few other bits and bobs around, or adjusting the amount of 'zap' the initiator provides, with typical yields up into 250-500 kilotons
Lastly, the H-bomb. This is the power of the stars, our Sun. Atoms of Hydrogen are fused together into Helium under cosmic heat and pressure, releasing; well, cosmic amounts of energy. This requires enormous energy to start the fusion reactions. The current H-bombs are all of the Teller-Ulam-Sakharov 'alarm clock' radiation implosion designs, where a tritium boosted fission core is used to provide the initial high energy and pressures to cause fusion in a secondary deuterium(heavy hydrogen) fusion fuel. The yield of these weapons is measured in equivalents of millions of tons of TNT, megatons. Typical yields are from 250 kilotons to 25 megatons. There is no theoretical limit on the number of secondary fusion stages that can be added to a H-bomb, except for the physical size of the package that your bomber or missile can carry, so no upper limit on the theoretical yield. The largest production weapons have been in the 25 megaton range. The experimental 'Tsar Bomba', a propaganda bomb, was designed as a three stage fission-fusion-fusion device with a theoretical yield of 100 megatons,though this was reduced to 50 megatons for the test, and even then, most of the released energy went straight up out of the atmosphere, which is why most modern H-bombs are of much lower yields, maxing out in the 20 megaton range.
Hope that helps.Its not an easy subject to compress into a few sentences, and I hope that's a reasonably concise description.
Happy Nuking!
