Just some small comments here. Yes, the question of whether or not Vassiliev invariants separate knots is still open.
One broader context for the question is to consider how Vassiliev invariants were first observed -- via the Vassiliev spectral sequence for the space of "long knots". These are knots of the form $\mathbb R \to \mathbb R^3$ which restrict to a standard (linear) inclusion outside of an interval, say $[-1,1]$.
There's a few standard ways in which the space of long knots can be turned into a topological monoid -- a "Moore loop space" construction which is pretty standard, or a "fat knots" construction from my "little cubes and long knots" paper (Topology 46 (2007) 1--27.). So this monoid operation is just the connect-sum operation, suitably jazzed-up to be strictly associative.
An observation for which I'm not sure if anyone has written up a complete proof yet is that the Goodwillie embedding calculus (an alternative approach to the Vassiliev spectral sequence, among other things) factors through the group-completion of the space of long knots. By this I mean, if K is the space of long knots, call the "group completion" the map $K \to \Omega BK$. $\Omega BK$ is where the "formal inverse" to a knot lives. The homotopy-type of $\Omega BK$ is computed in "little cubes and long knots". In particular, you can think of the Vassiliev spectral sequence as being an invariant of $\Omega BK$, not $K$. From the point of view of Vassiliev invariants this isn't such a major insight as $\pi_0 \Omega BK$ is the group-completion of the monoid $\pi_0 K$, so it's just a free abelian group on countably-many generators.
In particular, there are classes in $H^1 K$ for which there are no obvious finite-type approximations. $H_1 K$ has some torsion classes for which it's pretty unclear how to detect using the Vassiliev spectral sequence. Here is a mass of torsion computations for the homology of the long knot space: Ryan Budney, Frederick Cohen On the homology of the space of knots, Geometry & Topology 13 (2009) 99--139, arXiv:math/0504206.
edit: although "little cubes and long knots" describes the homotopy-type of the group completion of the space of long knots, it would be nice to have a more "concrete" embedding-space type description of the group completion. Mostovoy has an attempt in Short Ropes and Long Knots. I believe an idea like this should work and likely Mostovoy's idea works as well, but (again) this is something that needs to be written up for which there hasn't been enough time to start the project.
Probably a better answer will come along soon, but in the meantime...
I remember listening to a talk by biologists in the late 1980's about this very problem, but with knotted DNA. They used the Jones polynomial (or perhaps the Kauffman bracket polynomial), but perhaps because that invariant was trendy in those days.
I think professional knot enumerators first compute the Kauffman bracket (or some similar polynomial), and then count representations of the fundamental group into small symmetric groups. I've heard that in practice this almost always distinguishes distinct prime knots, but of course these are knots with relatively few crossings.
If you have hundreds of crossings, I think the most efficient thing to do would be to first find all Reidemeister 1 and 2 moves which reduce the number of crossings. Once that's done hopefully there are many fewer crossings and you can compute the Kauffman bracket using a straightforward algorithm.
Best Answer
I don't think that there has been a tremendous amount of progress in understanding the Kontsevich Invariant of a knot in the last decade or so. It appears that essential new ideas may be needed in order to answer the fundamental questions.
This is still more-or-less current, I think, and is a basic reference. Also T. Ohtsuki's Problems on invariants of knots and 3–manifolds.A more recent textbook reference is:
It contains a detailed discussion of the Kontsevich Invariant of a knot, including developments of the last decade such as work that has been done on the loop expansion and on the 2-loop polynomial in particular.With regard to clues for a final answer, not to toot my own horn, but Ohtsuki and I showed that 3 loop Vassiliev invariants do not distinguish a knot from its reverse (this is trivial for lower loop degree):
If this were true for general loop degree, then that would imply that there exist prime unoriented knots which cannot be distinguished by the Kontsevich invariant (this is a result of Kuperberg). My personal suspicion (without solid support) is that this is false for general loop degree, and indeed that the Kontsevich invariant does separate (unoriented) knot types.