what is the difference between a set's closure and completion?

In short, the answer to your question is "yes". To prove it we just need to prove that a closed subspace of a complete metric space is itself a complete metric space. For some background which may help in thinking about this topic, you can continue reading.

The completion and the notion of completeness can be formulated without an ambient space. For example, the completion of $\mathbb{Q}$ is always $\mathbb{R}$, regardless of whether we think of it as embedded in $\mathbb{R}$ or as by itself. (Here and throughout this post the metrics on $\mathbb{Q}$ and $\mathbb{R}$ are the Euclidean ones.)

The closure requires an ambient space. This is because the closed sets being intersected are actually closed in the ambient space, not necessarily in the space whose closure is being taken. If the ambient space is itself a complete metric space, the closure and the completion coincide. For example, the closure of $\mathbb{Q}$ in $\mathbb{R}$ is indeed $\mathbb{R}$. But if the ambient space is not a complete metric space then this is in general not true. Indeed, the closure of $\mathbb{Q}$ in $\mathbb{Q}$ is just $\mathbb{Q}$ (since $\mathbb{Q}$ is closed in itself).

No. The closure of a set happens inside a given space $X$. Completion on the other may require adding many new points also to the ambient space. For instance, take $X=\mathbb Q$ with its standard metric. The closure of $\mathbb Q$ is $\mathbb Q$, but the completion is $\mathbb R$.

Later edit, following adding the 'complete' to the question, the answer is 'yes'.