No, these are not the same thing. Formal schemes are to rigid-analytic spaces as $\mathbf{Z}_p$-schemes are to $\mathbf{Q}_p$-schemes.

The book *Lectures in Formal and Rigid Geometry* by Bosch is an excellent and friendly reference on this subject - take a look especially at sections 7.4 and 8.3.

In particular, let $K$ be a non-archimedean field (i.e. a field complete with respect to some $\mathbf{R}_{>0}$-valued multiplicative norm) and let $\mathscr{O}_K$ be its valuation ring. Then to any "reasonable" $\mathscr{O}_K$-formal scheme $\mathfrak{X}$, we can associate a rigid-analytic "generic fiber" $X = \mathfrak{X}_K$. (This is literally the generic fiber in the broader context of adic spaces, which subsume both formal schemes and rigid-analytic varieties).

We say that a formal scheme $\mathfrak{X}$ with $X = \mathfrak{X}_K$ is a *formal model* of $X$. It is a deep theorem of Raynaud that formal models of (reasonable) rigid-analytic spaces always exist, and are unique up to the operation of "admissible formal blowing up" (more precisely, the category of reasonable rigid-analytic spaces over $K$ is equivalent to the localization of the category of reasonable formal schemes over $\mathscr{O}_K$ with respect to this operation).

One warning: this "generic fiber" operation is *not* compatible with the usual one for schemes under analytification and formal completion.

For example, consider the affine $\mathbf{Z}_p$-line $\mathrm{Spec}(\mathbf{Z}_p[T])$. Its generic fiber is the affine $\mathbf{Q}_p$-line $\mathrm{Spec}(\mathbf{Q}_p[T])$. The analytification is the rigid-analytic affine line, which includes all elements of $\mathbf{Q}_p$ as $\mathbf{Q}_p$-points.

On the other hand, the formal completion of $\mathrm{Spec}(\mathbf{Z}_p[T])$ at $p$ is the formal unit ball $\mathrm{Spf}(\mathbf{Z}_p\{T\} := \varprojlim \mathbf{Z}/p^n[T])$. The generic fiber of this is the rigid-analytic unit ball, given by the max-spectrum of the ring $\mathbf{Q}_p\{T\} = \mathbf{Z}_p\{T\}[1/p]$. The $\mathbf{Q}_p$-points of this space are the elements of $\mathbf{Z}_p$.

**EDIT**

For completeness (i.e. in case my advisor is reading this), let me add a few details:

Let $\varpi \in \mathscr{O}_K$ be a pseudo-uniformizer, i.e. a non-zero element with $|\varpi| < 1$. We define rings of restricted power series over $\mathscr{O}_K$ to be
$$\mathscr{O}_K\{T_1, \ldots, T_n\} := \varprojlim_n (\mathscr{O}_K/\varpi^n)[T_1, \ldots, T_n] = \left\{\sum_\alpha a_\alpha T^\alpha \in \mathscr{O}_K[[T_1, \ldots, T_n]] \mid a_\alpha \rightarrow 0\right\}
$$
Then we define
$$
K\{T_1, \ldots, T_n\} := \mathscr{O}_K\{T_1, \ldots, T_n\}[1/\varpi] = \left\{ \sum_\alpha a_\alpha T^\alpha \in K[[T_1, \ldots, T_n]] \mid a_\alpha \rightarrow 0\right\}
$$

The "formal closed unit ball" in $n$ variables over $\mathscr{O}_K$ is $\mathrm{Spf} \mathscr{O}_K\{T_1, \ldots, T_n\}$. Note that it is the formal completion of affine $n$-space over $\mathscr{O}_K$ at the special fiber $\{\varpi = 0\}$.

A formal scheme $\mathfrak{X}$ over $\mathscr{O}_K$ is *admissible* if it is locally of the form $\mathrm{Spf} A$ where $A = \mathscr{O}_K\{T_1, \ldots, T_n\}/I$ with $I$ a finitely generated ideal, and $A$ has no $\varpi$-torsion. Essentially, this means that $\mathfrak{X}$ is locally the $\varpi$-adic completion of a *flat* finite-type $\mathscr{O}_K$-scheme.

The rigid-analytic closed unit ball in $n$ variables over $K$ is $\mathrm{Sp} K\{T_1, \ldots, T_n\}$ (as a set, it consists of maximal ideals in this ring). Rigid-analytic spaces are constructed by gluing together spaces of the form $\mathrm{Sp} (K\{T_1, \ldots, T_n\}/I)$, where $I$ is any ideal (automatically finitely generated and closed).

The generic fiber functor sends $\mathrm{Spf} (\mathscr{O}_K\{T_1, \ldots, T_n\}/I)$ to $\mathrm{Sp} (K\{T_1, \ldots, T_n\}/I)$. In particular, it sends the formal closed unit ball to the rigid-analytic closed unit ball. It extends to a functor from admissible formal schemes to rigid-analytic spaces.

Raynaud's theorem applies once we add additional (very mild) compactness assumptions on both sides: the rigid spaces must be "quasiseparated and quasi-paracompact", and the formal schemes must be "quasi-paracompact".

An admissible formal blowing up of an admissible formal scheme is a certain sort of blowup along a closed subset supported on the special fiber.