The "DNA computer" concept was created over a decade ago by Adlerman. As the article says, it's good for some calculations, not for all.
The demonstration I read in the 90's was about the "traveling salesman" problem: find the shortest route that passes through a set of cities, knowing the distance between them.
The idea is that you assign a segment of DNA to the trip between two cities, with a length proportional to the travel distance. You "shake" the molecules in a essay tube and make them bind. This way, you will get many DNA strands, each one representing a complete travel. You filter them and sort them: the shortest strand you find is your answer.
Now, DNA computing, even if faster than silicon based one for certain problems, it's restricted to problems that can be solved as I explain.
Quantum computing, on the other hand, could be a real breakthrough... In 2001 IBM factorized 15 (in 5 and 3) for the first time. It doesn't seem impressive until you realize that with some advances (well... perhaps a lot!
) eventually you could crack secure code, like RSA, some day.