Although the terms glue and adhesive are often used interchangeably I make definite distinctions.
PVA and other similar adhesives, Tite Bond, White glue, much of what is used in industrial applications that comes on 50 gallon drums, etc, I think of as flexible bonding agents.
Depending on the additives to the basic PVA chemistry they exhibit differing hardness, water resistance, heat resistance, etc. For example, Tite Bond comes in at least three separate formulas with different hardness, creep under load characteristics, open time and water resistance. This makes one type perhaps more suitable for specific piano related use than others.
Hide glue, which is a true glue, is rigid when set. Additives in the cold (liquid)versions might give you more open time at the trade off of less hardness and slightly lower strength.
Urea formaldehyde, is a synthetic resin made of a combination of urea and formaldehyde.
I think of it as a glue not an adhesive as it is brittle and inflexible like hide glue when cured.
It has the disadvantage of being non-clear (dark red in fact) and requiring a minimum wood moisture content of 8% to insure it doesn't form a starved glue joint. It is an innapropriate glue for those who make compression crowned sound boards as the EMC of the wood is too low.
Both PVA based adhesives and true glue require appropriate clamping pressure in order to produce a quality bond. Both require good mating of wood surfaces to prevent an excessive thickness of glue in the joint being bonded. I have often noticed thick glue lines in the spaces between go-bars on old removed sound boards where the ribs are easily dislodged with a sharp blow. This is one reason I prefer the use of Pnuematics when gluing ribs onto new panels. ( Not to mention the massive structure needed for a go-bar deck.)
Then there is the world of epoxy resins, an adhesive which can do so many tasks, requires the use of low clamping pressure, can bond items with poor mating, be modified for stiffness, sandabilty, open time, etc. Sorry for the run on sentences.
Of course, there are a myriad of other adhesives that we use daily. Contact cements, PVC-E, hot glue guns, etc.
Use them all, just know there characteristics.
Dale Fox RPT