To the best of my knowledge, there isn't any difference between the algorithmic methods and the NN methods. Those that can solve in polynomial time do not give a precise solution. Those that do give a precise solution do not solve in polynomial time. Of those that give a precise solution, the fastest takes 2^Nth$2^N$, but it blows up in terms of memory. The fastest good algorithm I believe is ConcordeConcorde.
The efficient algorithms solve in polynomial time, don't blow up in terms of memory, and give a solution close to perfect, say, within 2-3%. Again, to the best of my knowledge, no NN has beatbeaten the best algorithmic solutions, but there are suggestions that some NN solution could be faster.