One is a nuclear weapon in which the nuclear explosion is designed not to fully "burn" or spend all of the nuclear material. This leads to fallout that contains unspent radio active particles. These particles will typically have the half life of the material from which they are made. The effect of a weapon like this is to leave the area surrounding the blast contaminated. Contrast this to a neutron bomb which is very clean by comparison, emits a lot of radiation but leaves an area that can be re occupied in about two weeks.
The other dirty bomb is not a nuclear bomb at all. It is just radioactive material that is blown up by conventional weapons. The effect of this is a much smaller area is contaminated. It is un likely that a blast could be large enough to contaminate millions or hundreds of thousands. If wind conditions were right and the blast was large enough then the contamination could extend to thousands, and of course if it were detonated in a crowded city then thousands would be affected. The bigger danger is the spreading of the material via wind or water which would make containment and clean up tougher.
Uh, no.
Even the most "efficient" nuclear weapon consumes less than 1% of its fissile material. *Any* nuclear bomb is going to scatter the vast majority of its radioactive "fuel" over the countryside.
And while several pounds of vaporized Uranium or Plutonium scattered over hundreds of square miles would be nothing to look forward to, it wouldn't really be that much of a health hazard, overall.
What makes nuclear weapons a real mess from the standpoint of radiactive fallout is actually a side-effect from the explosion itself -- the high-energy neutron and photon spray of the blast itself causes the remnants of the bomb, and a lot of the matter around it (air, water, dirt, etc.) to undergo nuclear changes which cause *them* to mutate into radioactive material as well. For example, there's no Strontium or Iodine in the bomb itself, and not much in the environment, but the nuclear blast transmutates other elements in and near the blast into highly radioactive Strontium-90 and Iodine-131 (among others).
This is why airburst nuclear explosions create a hell of a lot less radioactive fallout than ground-burst explosions -- air transmutates into radioactive isotopes less readily than does the solid matter at ground level. And underwater nuclear explosions are really nasty -- the US tested underwater explosions in the "Crossroads" series of tests, and was astonished at how intensely radioactive the seawater around the tests had become.
A "dirty nuke" is not one that is "designed not to burn all its fuel", since no nuke does, instead it's one that is packed with material that is easily transmuted into nasty radioactive isotopes by the blast itself. Cobalt is one good candidate for such a "dirtifier".