Which kinds of closures for spice and herb containers exist and what are their advantages and disadvantages? I have an extensive collection of spices and dried herbs, which I store currently in apothecary bottles with a cork stopper.

Most websites that deal with closure types focus on the differences for beverages, especially wine. Considering the requirements for storing spices and dried herbs (protection from humidity, flavor-retaining, pest control and prevention of contamination by the storage container itself due to industrial binding agents within cork or PVC plasticizers), is a cork stopper the solution with the most benefits?
The only truly, definitely airtight seals widely available are either plastic snap locks with a gasket, or bail-lock type glass containers.

Mind that there are a few spices (clove powder and mixtures containing it) that can be surprisingly aggressive towards plastics.