(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.
Dear Kevin,
Here are some things that you know.
(1) Every non-tempered representation is a Langlands quotient of an induction of a non-tempered twist of a tempered rep'n on some Levi, and this description is canonical.
(2) Every tempered rep'n is a summand of the induction of a discrete series on some Levi.
(3) The discrete series for all groups were classified by Harish-Chandra.
Now Langlands's correspondence is (as you wrote) completely canonical: discrete series
with fixed inf. char. lie in a single packet, and the parameter is determined from the
inf. char. in a precise way.
All the summands of an induction of a discrete series rep'n are also declared to lie
in a single packet. So all packet structure comes from steps (1) and (2).
The correspondence is compatible in a standard way with twisting, and with parabolic induction.
So:
If we give ourselves the axioms that discrete series correspond to irred. parameters,
that the correspondence is compatible with twisting, that the correspondence is compatible
with parabolic induction, and that the correspondence is compatible with formation of
inf. chars., then putting it all together, it seems that we can determine step 1, then
2, then 3.
I don't know if this is what you would like, but it seems reasonable to me.
Why no need for epsilon-factor style complications: because there are no supercuspidals,
so everything reduces to discrete series, which from the point of view of packets are described by their inf. chars. In the p-adic world this is just false: all the supercuspidals are disc. series, they have nothing analogous (at least in any simple way) to an inf. char., and one has to somehow identify them --- hence epsilon factors to the rescue.
[Added: A colleague pointed out to me that the claim above (and also discussed below
in the exchange of comments with Victor Protsak) that the inf. char. serves to determine
a discrete series L-packet is not true in general. It is true if the group $G$ is semi-simple, or if the fundamental Cartan subgroups (those which are compact mod their centre) are connected. But in general one also needs a compatible choice of central character to determine the $L$-packet. In Langlands's general description of a discrete series parameter, their are two pieces of data: $\mu$ and $\lambda_0$. The former is giving the inf. char., and the latter the central char.]
Best Answer
As said in comments, the question has already been discussed on MO, see the links given there.
To summarize and complete, it is important to remember that in the case of number fields the correspondence between Galois representations and certain automorphic representations (the ones which are algebraic at infinity in the case of number fields) is only a part of the Langlands program, essentially because it concerns just certain automorphic representations, and Langlands functoriality is about all of them.
In the function field case, for $\operatorname{GL}_n$ over an arbitrary field, the correspondence has been completely done 15 years ago by Laurent Lafforgue. More recently (last year) Vincent Lafforgue (younger brother of Laurent) has done the sense Automorphic $\rightarrow$ Galois for an arbitrary reductive group (that is to an automorphic form he attaches a suitable $L$-parameter as conjectured, that is a kind of twisted Galas representation). In the introduction of his paper he seems optimistic about his prospect to solve (with Genestier) the converse direction, which entails carefully regrouping the automorphic representations into classes called $L$-packets.
In the number field case, much less is known, but much more than 20 years ago. In the sense Automorphic $\Rightarrow$ Galois, we can now do the case of automorphic representations for $\operatorname{GL}_n$ over a field with is either totally real or CM, and which are not only algebraic but regular at infinity. We can also do this for other groups (unitary, orthogonal, symplectic) but that gives no new Galois representations so I don't dwell on it (though in the proof, we need to do the case of these groups before going to $\operatorname{GL}_n$). The great final stone was put, after a huge collective effort leading to the case of self-dual or conjugate slef-dual representations, by Harris-Lan-Taylor-Thorne and then by another method by Scholze, which also deals with the case of torsion automorphic forms, not part of the initial Langlands program. The next main frontier seems to me to be able to deal with non-regular algebraic automorphic representations, the simplest case of which being algebraic Maass form for $\operatorname{GL}_2$.
Much progress has been made since Wiles and Taylor's proof of FLT on the converse direction Galois $\Rightarrow$ Automorphic, but it definitely lags behind the other sense. Essentially, due to work of Harris-Taylor and many others, the case of almost all conjugate self-dual Galois representation is done. Work in progress involving a numb or people could do many new cases, perhaps all where we know the automorphic to Galois sense.
This is essentially what is known about the Global Galois/Automorphic correspondence. For the functoriality for (non necessarily algebraic) automorphic representations, much less is known (Solvable base change for $\operatorname{GL}_n$ by Langlands and Arthur-Clozel, some transfer between classical groups, some low degree exterior powers) but much remains to be done.