In fact, Adleman solved a very small instance of a rather easy NP complete problem. 7 vertex Hamiltonian path is just too easy to bother with. Modern approximations routinely "solve" TSP with tens of cities.
Of course, we're not always sure we got an optimum (although branch and bound sometimes guarantees that, too). And DNA computing doesn't really scale. Adleman used 20-base oligomers; a bigger problem would require longer oligomers (length grows linearly with the size of the instance). And since "all" combinations are tried (in fact that's not true either, since there's nothing to guarantee this will happen for very large problems), exponentially many combinations would still have to happen to find the right one. And the number (read: amount) of DNA molecules produced would grow exponentially, too. (I don't mention error correction, which is also a real problem, as it is solvable).
So it's not clear that DNA computing is a real advance in solving hard problems. But it's neat as hell!