I am a beginning graduate student. I have the following basic question I am very confused about:

Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $p$ prime. Now take the ring $A$ of functions on $C$ regular away from a rational point, say $\infty$. Let $K$ be its function field and $K_\infty$ be the completed algebraic closure of $K$ wrt a valuation at $\infty$. Now define a rigid analytic function, say $f$, over $K_\infty$.

On the other hand suppose $B= \mathbb{F}_q[\theta]$, where $\theta$ is an indeterminate, and let $B_\infty$ be defined same way as $K_\infty$. Then define the Tate algebra to be $T_B = B_\infty[[T_1,\cdots, T_l]]$, where $T_i$ are indeterminates different from $\theta$. Again define the rigid analytic function, say $g$, over this Tate algebra $T_B$.

What is the difference (which one is more general) between theory developed around the functions $f$ and $g$?