I had always thought not, but I have seen enough people using them interchangeably (especially reffering to DNA as a polymer) to inspire doubt.
A polymer is a long chain made of of repeating segments of smaller molecules. Some proteins are polymers. DNA is a sort of polymer. Nearly all plastics are polymers, but no plastics are protiens. The words are not interchangeable.
Both proteins and DNA are instances of a general class of chemicals known collectively as polymers. PVC, or Poly-Vinyl Chloride, is an inorganic instance. There are literally millions of kinds of polymers in the world, most of the commercially important ones being either proteins or plastics.
So, what is a polymer? A polymer is a chain-like molecule, composed of simpler molecules (monomers) all strung together. Polymers perform many vital functions in our bodies, from storing genetic information (DNA and RNA) to actually doing the grunt work of metabolism and other cellular functions (proteins). The development of polymer theory played a huge role in our ability to create synthetic materials such as plastics, originally invented to mimic rubber and other natural polymers.
Well, DNA is not a protein, but it is a polymer. So is protein (a polymer, that is). A polymer is any long chain of smaller subunits. Consider glucose, which is a simple sugar. Put a long chain of them together and you get starch (a polysaccharide). Protein is a long chain of amino acids (a polypeptide). DNA is a long chain of nucleotides (a polynucleotide).
They are not synonomous. Proteins and DNA are types of polymer insofar as they are long chain molecules made of repeating subunits. However polymers are not just DNA and protein. Some purists will argue that DNA and most proteins are not true polymers because the individul subunits are not repeated regularly and some subunits are may not be repeated at all, however the purists are in the minority.
On preview I see QED bea me to it, but I’m posting anyway.
All proteins must be made up of segments joined by peptide bonds. All amino acids have two ends–one is the carboxylic acid end (which has a positive charge) and one is the amino end (which has the negative charge). A peptide bond is when the carboxy end of one amino acid is bound to the amino acid of another.
So proteins are a class of polymers, but not all polymers are joined in this way. As Blake says, some purists might object to proteins being called polymers, because rarely (and not that I know of) does a protein consist of all of one type of amino acid.
Proteins, carbohydrates, nucleic acids, and sometimes fats are sometimes called biopolymers to distinguish them from the large class of polymers.
How is PVC inorganic?
Quite a few ‘plastics’ don’t consist of only one type of monomer. The difference if that the monomers are repetaed in regular sequence. This occurs in some proteins such as keratin, but it;s unusual.
The word ‘biopolymers’ is used in the biological physics community and covers both proteins and DNA. I don’t know if this is used outside of this rather small niche.
When people refer to ‘organically grown vegetables’ I love to reply, ‘As opposed to aqueous?’
Many plastics are homopolymers, made from a single repeating unit, while most proteins are heteropolymers, made from several different compounds. Proteins all share a single type of bonding chemistry, the peptide bond, but other heteropolymers may contain several different types of linkages within a single molecule.
PVC is organic but not bio-organic, being a synthetic polymer.
Heh. And I thought PVC was inorganic because it contained chlorine.
Goes to show, even an incorrigible know-it-all can learn something. 
Hey easy e:
A minor point, but
is incorrect. Carboxyl=negative, amino=positive.
-Apoptosis