The neutrom bomb is known, quite ironically, as the "safe bomb", because there is, as such, no huge sod-off explosion.
Your post contains a lot of misinformation.
A neutron bomb is, to begin with, a nuclear fission bomb and functions as such. There
is a huge sod-off explosion -- not a multimegaton "destroy a large city plus its suburbs" fusion bomb explosion, but for most designs a classical multikiloton "destroy one town, or part of a large city" explosion.
All nuclear bombs generate a burst of free neutrons as a byproduct; a neutron bomb is built so as to enhance this to the point where the lethal radius of said neutron burst is significantly larger than that of the bomb's other effects. Therefore, the result if you were to drop such a bomb on a populated area would be just as devastating as if you dropped a conventional fission bomb, except that there would
also be a larger area outside of the devastation proper where people (and other lifeforms) suffered radiation poisoning (severity depending on exact distance, and degree of cover). In that larger area, you'd get the "people die or are incapacitated but buildings and roads aren't destroyed" effect.
The original idea for this type of bomb was, I believe, to use it as a tactical weapon against armor. A modern battle tank is fairly resistant to heat and shockwaves, so a tactical nuke has a fairly small kill radius against them; with a neutron bomb, you would kill or incapacitate the crew of a tank (or passengers in an APC, etc.) at much greater distance from the explosion, since the neutrons can penetrate a significant amount of armour (and those neutrons which are stopped will set off secondary radiation).
A neutron bomb would generate fallout to the same extent as a regular fission bomb would, detonated under the same conditions, i.e. highly variable. A bomb shelter protected under several meters of earth, rock or concrete would be safe (as long as it's not close enough to be blown up by the explosion), as even free neutrons can't penetrate solid objects
that well (and since they stop in the first meter or three of concrete, the secondary radiation they set off stops in the next meter or so).