(1) Regarding the relationship between geometric Langlands and function field Langlands:
typically research in geometric Langlands takes place in the context of rather restricted ramification (everywhere unramified, or perhaps Iwahori level structure at a finite number of points). There are investigations in some circumstances involving wild ramification (which is roughly the same thing as higher than Iwahori level), but I believe that there is not a definitive program in this direction at this stage.
Also, Lafforgue's result was about constructing Galois reps. attached to automorphic forms. Given this, the other direction (from Galois reps. to automorphic forms), follows immediatly, via
converse theorems, the theory of local constants, and Grothendieck's theory of $L$-functions in the function field setting.
On the other hand, much work in the geometric Langlands setting is about going from local systems (the geometric incarnation of an everywhere unramified Galois rep.) to automorphic sheaves (the geometric incarnation of an automorphic Hecke eigenform) --- e.g. the work of Gaitsgory, Mirkovic, and Vilonen in the $GL_n$ setting does this. I don't know how much is
known in the geometric setting about going backwards, from automorphic sheaves to local systems.
(2) Regarding the status of function field Langlands in general: it is important, and open, other than in the $GL_n$ case of Lafforgue, and various other special cases. (As in the number field setting, there are many special cases known, but these are far from the general problem of functoriality. Langlands writes in the notes on his collected works that "I do not believe that much has yet been done beyond the group $GL(n)$''.) Langlands has initiated a program called ``Beyond endoscopy'' to approach the general question of functoriality. In the number field case, it seems to rely on unknown (and seemingly out of reach) problems of analytic number theory, but in the function field case there is some chance to approach these questions geometrically instead. This is a subject of ongoing research.
I think it is easiest to illustrate the role of the Langlands program (i.e. non-abelian class field theory) in answering this question by giving an example.
E.g. consider the Hilbert class field $K$ of $F := {\mathbb Q}(\sqrt{-23})$; this is a degree 3 abelian extension of $F$, and an $S_3$ extension of $\mathbb Q$. (It is the splitting field of the polynomial $x^3 - x - 1$.)
The 2-dimensional representation of $S_3$ thus gives a representation
$\rho:Gal(K/{\mathbb Q}) \hookrightarrow GL_2({\mathbb Q}).$
A prime $p$ splits in $K$ if and only if $Frob_p$ is the trivial conjugacy class
in $Gal(K{\mathbb Q})$, if and only if $\rho(Frob_p)$ is the identity matrix, if and only
if trace $\rho(Frob_p) = 2$. (EDIT: While $Frob_p$ is a 2-cycle, resp. 3-cycle, if and only if $\rho(Frob_p)$ has trace 0, resp. -1.)
Now we have the following reciprocity law for $\rho$: there is a modular form $f(q)$, in fact
a Hecke eigenform, of weight 1 and level 23, whose $p$th Hecke eigenvalue gives
the trace of $\rho(Frob_p)$. (This is due to Hecke; the reason that Hecke could handle
this case is that $\rho$ embeds $Gal(K/{\mathbb Q})$ as a dihedral
subgroup of $GL_2$, and so $\rho$ is in fact induced from an abelian character of the
index two subgroup $Gal(K/F)$.)
In this particular case, we have the following explicit formula:
$$f(q) = q \prod_{n=1}^{\infty}(1-q^n)(1-q^{23 n}).$$
If we expand out this product as $f(q) = \sum_{n = 1}^{\infty}a_n q^n,$
then we find that $trace \rho(Frob_p) = a_p$ (for $p \neq 23$),
and in particular, $p$ splits completely in $K$ if and only if $a_p = 2$.
(For example, you can check this way that the smallest split prime is $p = 59$;
this is related to the fact that $59 = 6^2 + 23 \cdot 1^2$.).
(EDIT: While $Frob_p$ has order $2$, resp. 3, if and only if $a_p =0$, resp. $-1$.)
So we obtain a description of the set of primes that split in $K$ in terms of
the modular form $f(q)$, or more precisely its Hecke eigenvalues (or what amounts
to the same thing, its $q$-expansion).
The Langlands program asserts that an analogous statement is true for any
Galois extension of number fields $E/F$ when one is given a continuous
representation $Gal(E/F) \hookrightarrow GL\_n(\mathbb C).$ This is known
when $n = 2$ and either the image of $\rho$ is solvable (Langlands--Tunnell) or $F = \mathbb Q$ and $\rho(\text{complex conjugation})$ is non-scalar (Khare--Wintenberger--Kisin).
In most other contexts it remains open.
Best Answer
This question has already been discussed here, though perhaps not exactly on these terms (but see What makes Langlands for n=2 easier than Langlands for n>2?).
So first there is an ambiguity of what id global Langlands in general. Langlands was interested in all cuspidal automorphic representations for $Gl_2$: he conjectured they should be in natural bijection with complex two-dimensional representations of a large group called the Langlands group of $\mathbb Q$. Problem: the Langlands group of $\mathbb Q$ is a conjectural object, whose very existence depends on a hypothetical structure of Tanakian category on the category of all cuspidal automorphic representations for $Gl_n$ when $n$ varies. For this reason, it does not really makes sense of Langlands' conjecture for $Gl_2$: it is really a conjecture concerning all $Gl_n$ together.
Therefore, when people refers to global Langlands for $Gl_2$ they mean a more restrictive conjecture which deal with only a subclass of the set of cuspidal automorphic representation, namely the one which are algebraic at infinity. The conjecture is then as follows:
Conjecture: Fix an auxiliary prime $\ell$. There is a natural bijection between
(1) the set of algebraic at infinity cuspidal automorphic representations for $Gl_2$
(2) the set of 2-dimensional continuous irreducible Galois representation of $G_{\mathbb Q}$ over the algebraic closure of $ \mathbb Q_\ell$ which are unramified at almost all primes and de Rham at $\ell$.
Moreover, this bijection should be compatible to the local Langlands correspondence at almost every prime $p \neq \ell$ (or better, at every place). (This condition fixes the bijection uniquely).
So what is proved in this conjecture. Well there are three type of representation $\pi_\infty$ which are algebraic, which leads to a tripartition of the set (1): (1a) : the set of cuspidal automorphic forms for $Gl_2$ for which $\pi_\infty$ is a discrete series which is essentially the set of new cuspidal modular forms of weight $k\geq 2$. (1b) : the set of cuspidal automorphic forms for which $\pi_\infty$ is a limit of discrete series, which is essentially the set of new cuspidal modular forms of weight $k=1$. (1c) : the set of cuspidal automorphic forms for which $\pi_\infty$ is a specific principal series, which corresponds to the set of Mass form with Eigenvalue for the Laplace-beltrami operator (a.k.a. the Casimir) 1/4.
To prove the conjecture, one needs (a) to construct one map from (1) to (2) which is compatible with Local Langlands, and (b) prove that it is surjective (the injectivity will be easy by strong multiplicity 1 for $Gl_2$). Now this map is known only in the case (1a) and (1b), not at all in the case (1c). Worse: the map was already constructed in 1970 in the cases (1a) and (1b) by Deligne and Deligne-Serre, and we have made no progress since. So it is very hard to say how far we are from constructing the map in general. There is no publicly voiced ideas to prove it, but maybe some one will come tomorrow and prove it. What about subjectivity? Well, the image of (1a) and (1b) together should be (2ab), the set of Galois rep. as above which in a edition are odd. the image of (1a) should be the set (2a) of odd representations with distinct Hodge-Tate weights, and actually we know that there is a surjection (1a) to (2a): this is the part the conjecture of Fontaine-Mazur proved by Kisin and Emerton. We don't know yet the subjectivity of (1b) to (2b) (defined as (2ab)-(2a)), which is essentially Artin's conjecture but special cases have been done by Taylor, Buzzard, and Calegari.