enzyme is defined as a catalytic protein, meaning a protein which
catalyzes a chemical reaction. For example, carbonic anhydrase is an
enzyme that catalyzes (i.e. lowers the activation energy of) the
reaction CO2 + H2O <-> H2CO3. However, not all proteins catalyze
chemical reactions. Some have structural or mechanical roles, like
microtubules in the cytoskeleton or myosin in muscle. Others are used
for transmitting signals, like insulin, a peptide hormone. These are
all proteins, but not enzymes.
FYI, it's not such a clear distinction any more. With the discovery of
catalytic RNA (ribozymes) and other such non-protein catalysts, as well
as expanding evidence of carbohydrate- or lipid-modified proteins, it's
more a question of terminology whether all enzymes are proteins or not.