What is the difference between polymer science, polymer chemistry, and polymer physics?

Polymer Science is a generic term covering the various subdivisions of the subject.

Polymer chemists are mostly interested in the task of making plastic substances from their precursor chemicals; by reacting ethylene, propene or other hydrocarbons with catalysts under appropriate conditions Eg high pressure. Plastics or thermoplastics are typically made by free radical chain reactions. In contrast polymer resins or thermosets, another class of man-made polymers, are made by cross linking various reactive momomers or oligomers together, Eg polyurethane paint when spread into a thin film eg on a wall, undergoes a cross linking reaction to form a resin; the crosslinking reaction is induced by atmospheric oxygen. Resins unlike plastics are essentially one big molecule! That's why there are few if any solvents for them. In contrast plastics can be softened with solvents being linear in their structure at the molecular level.

The distribution of polymer chain lengths in a given plastic material is largely controlled by the kinetics of the free radical chain reaction, in the monomer to polymer conversion process. So you've got to like reaction kinetics if you want to be a polymer chemist!  Lots of physical chemistry.
 Rubbers are amorphous thermoplastics above their glass transition temperature. Polymer spectroscopy carried out for polymer material identification purposes is also a speciality within the field of polymer chemistry. Also gel permeation chromatography or gpc which is another polymer characterization technique.

In contrast, polymer physicists are more interested in the physical properties of the materials polymer chemists have made for us. Such as the optical properties of the polymer ( Eg contact lenses ), the behavior of polymers in solution ( Eg non drip paint; which is a non-Newtonian fluid ie its viscosity decreases when a shearing force is applied to it eg  paint brush spreading paint on a wall, and increases when that force is removed, Eg paint brush removed from wall on its way to paint tin for more paint ). Polymer physics also includes the study of light scattering by polymers;( eg used as a polymer characterization method );  the study of the statistical mechanics of rubber elasticity, plus other theoretical topics. Also polymer crystallography, and Raman spectroscopy.

Polymer material scientists tend to be interested in the engineering properties of substances  that happen to be a plastic or resin. So like metallurgy, polymer materials science concerns itself with stress strain relationships & the tensile strength of the material; with the toughness of the polymer material; its hardness; the material's melting point or glass transition temperature; creep; & materials selection for mechanical design. Plus properties metals don't have such as viscoelasticity.

The strength of materials is a fundamental subject within materials science whether the material under study is a metal, plastic, resin, composite, glass, a ceramic, or concrete; or a natural material such as wood or granite.

Polymer blending is another subject within polymer materials science. Also thermal analysis, using techniques such as TG, DTA, & DSC.
 Turning to polymer engineering. Polymer engineers often have degrees in mechanical engineering. Polymer engineering concerns itself with the moulding or castings of polymer materials into useful engineering products. Things such as plastic buckets, rainwater goods, tyres, yacht hulls, plastic bottles, aircraft windows,  plastic shopping bags, sneakers, shoes, car panels, train seats, aircraft interiors, vending machine tea cups, toys, sports equipment etc, the list goes on !
  Major polymer engineering processes deployed to make such goods include; plastic injection moulding ( plastic washing up bowls made using this type of moulding machine ), vacuum forming ( plastic egg cartons made in this way ), extrusion machines  ( PVC drain pipes ), plastic composite mouldings ( yacht hulls, aircraft wings, military technology  ), reaction injection moulding  ( automobile panels ), blow moulding  ( plastic bottles ), & rotomoulding  ( large plastic tanks ).
Polymer technology has a slightly broader remit covering subjects in addition to polymer engineering ranging from polymer additives for car engine oil ( Eg star polymers ), to medical polymers (Eg pace makers ), polymers in electronics, household paints, fabrics, rope, polythene beakers for hydrofluoric acid. In fact any useful application.