TITLE: Formal schemes vs formal power series
QUESTION [5 upvotes]: Take $X = \mathbb{A}^1$ and $Y = \{0\}$.  I want to take the formal group scheme at $Y \subset X$.  This is a locally ringed space, $(Y, \mathcal{O}_{
\hat{X}})$ where $\mathcal{O}_{\hat{X}}$ is the $(x)$-adic completion of $k[x]$, i.e. $k[[x]]$.
This might be vague, but why formal schemes, i.e. what is the difference between this formal scheme and $Spec (k[[x]])$?  For example, is the category of coherent sheaves (defined on any ringed space) equivalent for the two?  I suppose the underlying topological space is different, but why would you prefer one over the other?

REPLY [3 votes]: Here's an answer that's somewhat what I'm looking for.  First, it seems that they have equivalent categories of coherent sheaves, given by Lemma 2.2 in the following reference.
However there is a difference in the functor of points, namely, considering $\hat{\mathbb{G}}_a(R)$ as a direct limit, one wants
$$\lim Hom(k[t]/t^i, R)$$
which is exactly the set of nilpotent elements of $R$.  In particular, $k[[t]]$ has no nilpotent elements, but
$$Hom(k[[t]], k[[t]])$$
is nonempty.