My own interest in automatic groups has been principally algorithmic, and I believe that this was Thurston's original motivation for studying them - they provided a method for carrying out practical computations in a variety of interesting groups with negative curvature. Once a (geodesic) automatic structure has been computed, you can compute the growth function of the group (this was of particular interest to Thurston), you can reduce words to normal form rapidly, you can usually compute the orders of elements, you can solve the membership problem for quasiconvex subgroups, and so on.
It is true that research into the theory of automatic groups has to some extent ground to a halt, because the remaining open problems seem very hard. For example, there are very few techniques for proving that a group is not automatic, particularly if it has quadratic Dehn function. Although nobody seems to believe that all automatic groups are biautomatic, people seem to have given up on trying to find an example.
But, for a simple computational group theorist like myself, the wonderful thing is that, if you are given a group defined by a finite presentation, then you do not need to know in advance whether the group defined is automatic. You can just run the programs and try and prove that it is. Informally, this results from the nice property of finite state automata, that you can often construct other automata that prove that your original automata do what they are supposed to do - in this case, prove that they define an automatic structure of the group. Of course, for many groups (such as non-automatic groups!) this won't work, but it usually becomes clear very quickly if the programs are not going to work, because the fellow-travelling property of automatic groups appears not to hold.
There have been several examples of groups for which it was not known whether they were finite or infinite, which were proved infinite using the automatic groups programs. One such was the Heineken group defined by the presentation
$$\langle x,y,z | [x,[x,y]]=z, [y,[y,z]]=x, [z,[z,x]]=y \rangle$$
which had been open for many years, as a candidate for a finite group with a balanced presentation. It turned out that it was infinite, and word-hyperbolic. (Incidentally, this seems to me to be a possible counterexample to the suggestion that all hyperbolic groups might be residually finite, but I have no idea how to go about investigating that.)
Other examples are proofs that some of the groups in families defined by Coxeter are infinite, such as members of the family
$$(l,m,n;q) = \langle x,y \mid x^l=y^m=(xy)^n=[x,y]^q=1 \rangle.$$
A year or two ago, with the help of a large and difficult computation, we managed to prove that $(3,5,7;2)$ is automatic and infinite. There are now only three groups in this family that remain to be resolved.