Question: SRBs and missiles use [grain](https://space.stackexchange.com/questions/14107/ways-to-obtain-thrust-curves-of-different-grain-geometries) to regulate thrust over time, as only exposed surface of the propellant burns.

But what causes propellant to burn only on the surface, and regulates the speed at which the surface burns? Surely not atmospheric oxygen supply (which is what causes similar property of free-burning solids) as it has oxidizer distributed throughout the volume; there's no apparently clear reason why the solid fuel couldn't just burn all at once throughout the volume; something puts it apart from common explosives which are effectively just that, a kind of fuel and oxidizer mix that burns all at once all throughout the volume, in enclosed space, releasing all the combustion products at once.

So - what chemical additives or properties set solid propellants apart from explosives?

Answer: 
A propellant deflagrates, rather than explodes. The difference between them is that deflagration relies on the thermal energy of the flame front, rather than the energy of the shockwave caused by detonation.

The key to this process is a [balance](https://en.wikipedia.org/wiki/Deflagration#Flame_physics) between two factors, the activation energy of the propellant and the temperature at which the propellant burns. This balance is set against the thermal diffusion of the propellant. If the heat cannot diffuse fast enough into the propellant, it won't stay lit. If it diffuses too fast... well... you wondered what keeps them from going boom?
Looking through the description of [APCP](https://en.wikipedia.org/wiki/Ammonium_perchlorate_composite_propellant), I noticed that it contains metal *powder*. That means your assumption about the oxidizer isn't correct. At microscopic level, the oxidizer is *not* distributed throughout the volume of the metal. The distance between oxidizer molecules and metal atoms is measured in micrometers, not nanometers. Of course, a micrometer still isn't that much, which explains why the reaction is very quick once the propellant is vaporized. The atoms only need to travel those few micrometers before they can react. But that's not instant, and it limits the burn rate.