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?
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.