What is the purpose of a honey dipper? Why not just use a spoon? Is
there some added advantage?

If you're speaking of the kind of honey dipper that has grooves in it,
its advantage over a spoon is that it is less likely to drip. The
honey wraps around it as you twirl it, and it's less messy than using
a spoon.

I wonder if it might also have something to do with the antibacterial
properties of wood...

Now let us consider, not a honey dipper, but rather, a honey *wagon*.

   The purpose of a honey wagon is to sell those bacon-lettuce-tomato
sandwiches, which are strangely delicious, despite the toasted bread
being conspicuously soggy, and being entombed within a nearly
impenetrable barrier of convoluted handiwrap. A side purpose is to
sell cans of Coke which are invariably not *quite* cold enough.

Two words - surface tension.  The grooves allow the honey (with a high
surface tension) to be pulled in while they are horizontal, but flow
out when they are vertical.  I've seen both plastic and wood, so I
would guess there is no particular advantage to the material, other
that what is cheap to make.

The purpose of the wood over metal and plastic is because both metal
and plastic will degrade slightly while in the honey, this degrade
causes either a metalic tast and off coloration to the honey in the
case of metal. Or a slight discoloration of the honey with some
plastics, also many plastics are not as durable over time.
Also being wood it is more porus and will take on the flavor of the
honey instead of imparting its own flavor to the honey. Wood lasts
longer and does not chemically harm the honey in any way, this is why
wooden honey dippers have been used for generations.
Modern plastics do not degrade as much as older plastics did and most
likely would not cause a problem with the honey. But when honey
dippers were first used the only alternative was metals that would
change the taste and character of the honey.