Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects) The first one $\mathrm{Spec}\mathbb{C}[[t]]$ is a scheme, the second one $\mathrm{Spf}\mathbb{C}[[t]]$ is a formal scheme. In my mind they both realize an "infinite order infinitesimal neighbourhood of a point in $\mathbb{A}^1_{\mathbb{C}}$ ". The "formal disc" is an increasing limit of finite order infinitesimal neighbourhoods; pretty much like an increasing union of closed subschemes that are thickenings of the reduced point. How should I think of the others? So

What's the difference between these objects, intuitively? How are these objects related to the fact that $\mathbb{C}[[t]]$ actually carries an adic topology? What is each of them best suitable for? When to "use" one and when the other(s)? 

Of course the same questions could be asked replacing $\mathbb{C}[[t]]$ with any complete local ring $A=\hat A$.
 A: One good way of seeing the difference between $\mathrm{Spec}\,\mathbb{C}[[t]]$ and $\mathrm{Spf}\,\mathbb{C}[[t]]$ is to look at what functors they represent on affine schemes. In fact we have
$$\mathrm{Hom}(\mathrm{Spec}\,A,\mathrm{Spec}\,\mathbb{C}[[t]])=\mathrm{Hom}(\mathbb{C}[[t]],A)$$
This is just the ordinary morphisms of rings, and it's not particularly different from any map from a DVR into $A$.
$$\mathrm{Hom}(\mathrm{Spec}\,A,\mathrm{Spf}\,\mathbb{C}[[t]])=\mathrm{colim}_n\mathrm{Hom}(\mathbb{C}[t]/(t^n),A)=\{a\in A\mid a\textrm{ nilpotent }\}$$
More generally, when instead of $\mathrm{Spec}\,A$ we put any scheme $X$, we get the global sections of $X$ that are locally nilpotent (i.e. nilpotent when restricted on affine schemes).
In this sense you can think of $\mathrm{Spf}\,\mathbb{C}[[t]]$ as the completion of $\mathrm{Spec}\,\mathbb{C}[t]$ at the origin: maps into $\mathrm{Spf}\,\mathbb{C}[[t]]$ are the same thing as maps into $\mathrm{Spec}\,\mathbb{C}[t]$ whose set-theoretic image is (contained in) the origin.
Note that this works for all affine formal schemes: if $R$ is a ring and $I$ an ideal, $\mathrm{Spf}\,R^\wedge_I$ represents the functor sending $X$ to the maps of schemes $X\to \mathrm{Spec}\,R$ such that the set-theoretic image is contained in $V(I)$.
A: To complement the other answers, I would like to add a word on the analytic spectrum $\mathrm{Specan}(\mathbb{C}[[t]])$. 
First, let me say that I am not sure what $\mathrm{Specan}$ means and have no idea where the notation comes from. On the other hand, it makes sense to consider the analytic spectrum of $\mathbb{C}[[t]]$ in R. Huber's theory of adic analytic spaces (where it would be called $\mathrm{Spa}(\mathbb{C}[[t]],\mathbb{C}[[t]])$) and in V. Berkovich's theory of analytic spaces (where it would be called $\mathcal{M}(\mathbb{C}[[t]])$ ; one would also need to prescribe the absolute of $t$, say $|t| = r \in (0,1)$, because the theory requires actual absolute values and not merely equivalence classes, i.e. valuations, but this is not really an issue).
Let me start with adic spaces. In this case, the spectrum is made of one closed point ($t=0$) and one open point (associated to the $t$-adic valuation). This open point is really the generic point of the space. More generally, you can associate (fully faithfully) an adic space to any sufficiently nice formal scheme and take its generic fiber inside the category of adic spaces. So here you really have the formal scheme again but, in the underlying space, you see its special fiber and its generic fiber (whereas $\mathrm{Spf}(\mathbb{C}[[t]])$ only shows the special fiber).  
(Everything here looks quite similar to $\mathrm{Spec}(\mathbb{C}[[t]])$, so one may wonder why bother with formal schemes, fancy analytic spaces, etc. This is a only because the chosen situation is very simple (affine in particular) and you will have a hard time representing infinitesimal neighbourhoods by algebraic objects very quickly as soon as you start glueing.)
The situation with Berkovich spaces is very similar except that you will see a segment $[0,r]$ instead of two points. The point $0$ corresponds to $t=0$ and the other points all correspond to $t$-adic absolute values with different normalizations (given by $|t| = s$ for $s\in (0,r]$). Here again, you see a special fiber and a generic fiber (and even several equivalent copies of it). Note that the underlying space is Hausdorff and compact. This is a general feature of Berkovich spaces which is quite nice, although it is not clear how useful it would be in this particular situation. 
A: In a nutshell:


*

*$\mathrm{Spec}~\mathbb{C}[[t]]$ Is a “trait” i.e. the spectrum of a discrete valuation domain, with a generic (open) point and a closed point. It might be imagined as a refinement of the usual algebraic germ of the affine line at the origin, namely $\mathrm{Spec}~\mathbb{C}[t]_{\langle t \rangle}$.

*$\mathrm{Spf}~\mathbb{C}[[t]]$ is the completion of $\mathbb{A}^1_{\mathbb{C}}$ along the origin. It is interpreted as the union of all the infinitesimal neighborhoods of the origin in the line. As such, is strictly contained in any space of germs: there is a chain of canonical morphisms 
$$\mathrm{Spf}~\mathbb{C}[[t]] \to \mathrm{Spec}~\mathbb{C}[[t]] \to \mathrm{Spec}~\mathbb{C}[t]_{\langle t \rangle}$$

*$\mathrm{Specan}~\mathbb{C}\{t\}$ where $\mathbb{C}\{t\}$ denotes the local ring of convergent power series at the origin represents the analytic germ of $\mathbb{C}$ at the origin, usually denoted $(\mathbb{C}, 0)$. It also sits between $\mathrm{Spf}~\mathbb{C}[[t]]$ and $\mathrm{Spec}~\mathbb{C}[t]_{\langle t \rangle}$.


The topology in the sheaf of rings of $\mathrm{Spf}~\mathbb{C}[[t]]$ is responsible for its hidden part. If instead of $\mathbb{C}$ one takes an ultrametric field, then the chimeric “generic fiber” would be a model for the rigid analytic closed disk. For $\mathbb{C}$ one does not have something like the complete valuation subring playing the role of $p$-adic integers for $\mathbb{Q}_p$.
