There is no difference between a metrizable space and a metric space (proof included). Willard says, "Whenever $(X,\tau)$ is a topological space whose topology $\tau$ is the metric topology $\tau_{\rho}$ for some metric $\rho$ on $X$, we call $(X,\tau)$ a metrizable topological space."
I think giving a proof is the best way to illustrate how I think of these concepts and illustrate the exact points that I am not understanding.
Theorem: Metric space iff metrizable space.
(->) Let $(X,\rho)$ be a metric space. Consider the topology generated by this metric, $\tau_{\rho}$.  Then $(X,\tau_{\rho})$ is a topological space whose topology is the metric topology for some metric, so by definition, metrizable.
(<-) Let $(X,\tau)$ be a metrizable space.  There $\exists \rho$ a metric such that $\tau$ is the metric topology given by $\rho$.  And so $(X,\rho)$ is a metric space.

A metric space is an ordered pair $\langle X,d\rangle$ such that $X$ is a set and $d$ is a metric on $X$. A metrizable space is an ordered pair $\langle X,\tau\rangle$ such that $X$ is a set, $\tau$ is a topology on $X$, and there exists a metric on $X$ that generates the topology $\tau$. These are clearly not the same thing. A metric space has a specified metric, and the topology, though definable from the metric, is unspecified; a metrizable space has a specific topology, and while it is possible to define metrics that generate that topology, none is specified.
