Adhesives (glue in common parlance) serve several functions in making beehives. They increase the stability and strength of joints and also fill the minute cracks in joints that would otherwise be places where bugs and bacteria may lurk.
When I first started making beekeeping equipment I used a cold water casein resin glue known as 'Cascamite' This achieves a very high joint strength, but the constant wetting and drying that occurs with beehives causes expansion and contraction of the joints which in turn causes the joints to separate as the glue itself is rigid and will not flex. Frost or freezing will also disrupt the joint.
Waterproof PVA Glue was suggested by a chemist that I knew, who worked for an adhesive company, I replied that I thought PVA was water soluble and was duly treated to a lecture on various latex additives that migrate to the surface of exposed glue and form a waterproof seal. This gives a joint that will move with expansion without the glue itself 'washing away' by the action of rain. The glue within the sealed joint is able to exchange moisture with the wood and thus expand and contract in sympathy.
I have been using this type of adhesive now for something like twenty five years, it is easy to clean from a freshly made joint with a damp cloth, it has hardly any smell, but it is of adequate strength.
One slight disadvantage of this type of PVA glue, is that maximum strength is not developed for about 24 hours so there are often many items in cramps (clamps) in my workshop while they dry. I do not see this as a problem as most other types of glued joint will require to be cramped up for a similar time.
Building Adhesive (the type that indicates that no nails are required) gives a strong and waterproof joint. I have no personal experience of using this material in the assembly of wooden beekeeping equipment, but I have used it for DIY around the house and would say that the joints made with it are stronger than the wood itself. This 'no nails' glue can be used to good effect in making wooden feeders.
Gorilla Glue is a polyurethane adhesive that is very useful glue in many other circumstances than just beekeeping or woodworking, it is expensive, but a little goes a long way as it foams and expands as it cures.
Cured Gorilla glue can only be removed with chisel, scraper, or sand paper.
I think that I would have used Gorilla glue myself, for making beekeeping equipment and many other purposes, had I discovered it earlier... As it seems to be a glue that is suitable for almost all purposes..