What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
Expanding a little on what Arkamis said ...
Some people would define a spline to be any piecewise polynomial function. For example, deBoor's book uses this definition, and it's one of the definitive works on the subject. With that definition, there is no difference between the two kinds of interpolation you mentioned, of course.
Other people (like here) define a spline of degree $m$ to be a piecewise polynomial with $m-1$ continuous derivatives. So, with this definition, cubic splines (degree 3) would be $C_2$. These folks sometimes refer to less continuous piecewise polynomials as "subsplines" or "defective splines". In fact, a spline of degree $m$ that has a continuous $(m-k)$-th derivative is said to have defect $k$.
It depends on your piecewise polynomial specification.
By definition, a spline is a "sufficiently smooth" piecewise polynomial interpolant. The "sufficiently smooth" part comes from mandating how many orders of derivatives must be equal between successive splines.