A siege takes place at the defensive line and storming breaks through the defensive line. Assuming you refer to a castle, then the defensive line is the curtain wall or other fortification. Once a siege has broken through fortifications, the combatants are said to be storming.

To add to r/idjet's answer, a "storming" (generally of a fortification or a castle) was an assault or attack, intended to get over the walls (with ladders, ramps, or a siege tower), or through the gate (with a battering ram or a bombard) or through a "breach" in the walls (made by a mine or by "siege engines" or by cannon).

A siege, on the other hand, need not involve an attack on the fortification (though sieges often were accompanied by or included assaults or attempted assaults). A siege, is essentially a blockade. The fortification is surrounded. The entry of re-supply is prevented. The garrison is cut off. Eventually (if the siege persists) they will become so starved and weakened that they will subcome to assault, or starve, or surrender.

It was difficult to take a Medieval Castle by storming. Small numbers of defenders could often hold the castle against overwhelming enemy forces.

All fortifications, however, would eventually be defeated by an effective siege. If the fortress was well supplied, however, it could hold out against siege for a long time, and hope that allies would come to break the siege, or that the attackers would need their army elsewhere, or that the attackers would be devastated by disease or would themselves deplete the surrounding countryside of food and be forced to move on to re-supply themselves.