Most spicy food we eat gets its spice from capsaicin, which is in peppers. FOr peppers its a defense against animals. Capsaicin is a stable alkaloid at room temp and hydrophobic, so when its in your mouth it only spreads with saliva.
Horseradish is different. Horseradish, mustard, and wasabi produce 2 agents that are harmless: a chemical called sinigrin, and an enzyme myrosinase. They're separate in the plant but when the plant is damaged (ie chewed) they mix. The result of the mix is mustard oil.
Mustard oil gets in the mucous membranes, which react in the same way they react to capsaicin. But at the same time, it interacts with cells in your mucous membranes differently thats why it acts as in irritant.
Plus mustard oil is volatile at room temp, and evaporate in the heat of your mouth. So when it evaporates in your mouth, it spreads through your membranes. But its also why it dissipates quickly as opposed to eating a hit pepper and it stays with you.
Side note, thats also why mustard gas stings your face and sinuses. So eating horseradish is sort of like a mini dose of mustard gas.
