While it's correct to say these terms can be used interchangeably they should not, the meanings when used technically are quite distinct.
The problem begins with the word drying being overused in product descriptions for paints and other coatings; that is, "drying" is used loosely and therefore will always be inaccurate in some instances. But the concept of paint or varnish "drying" as a catch-all for any process of hardening is far too entrenched to change.
Drying literally means drying, an actual drying process where at the simplest water is lost by evaporation. For other coatings it is merely a different solvent (or mixture of solvents) that is evaporating.
Curing on the other hand is less of a populist term and should have its correct use maintained so as not to muddy the understanding of its meaning.
Most paints and varnishes, as well as many lacquers, undergo both drying and curing processes. Drying is the initial phase, where the coating shrinks due to the loss of the solvent component. Curing is the second (usually much longer) phase where the coating changes physically and/or chemically; it may swell slightly during this process.
Where there is a chemical change during curing it can be components within the coating combining and reacting, or a reaction with oxygen (or water) from the atmosphere. An example of the latter occurs with oils such as linseed oil and tung oil, which undergo oxidative polymerisation, changing from oils into a type of natural polymer by reacting with oxygen.
Where a coating only has a drying process, such as with watercolour paint or shellac, it is chemically unchanged and therefore remains soluble in the original solvent. This is why even decades later watercolours are water-soluble, and why shellac can be re-dissolved in alcohol.
When a coating cures its chemistry changes from that of its original liquid form, so that for example a paint that was soluble in mineral spirits (white spirit in the UK) at the time of application will, after curing, no longer be soluble in it.