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][1]* 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 $\mathbf{Z}_p$. 


  [1]: https://www.springer.com/gp/book/9783319044163