Basically, a conventional bomb uses a chemical explosive as the
source of its destructive power. A nuclear weapon uses nuclear
material to create an explosion. A nuclear explosion is much
larger, and also emits ionizing radiation. A chemical weapon does
not emit any radiation.
A nuclear weapon's yield is measured in Kilotons (thousand
tons). In very simplified terms, this means that a nuclear weapon
with a 475 kiloton yield produces an explosion comparable to
475,000 tons of TNT (TNT is a chemical explosive). That's A LOT of
TNT and it would take up a bit of space. A nuclear weapon with this
yield may only be a few feet long and a foot wide, and the actual
nuclear material may be the size of a grapefruit.