In operations research, it's common to stop when a solution is provably within 1e-4 of optimum. Off the top of my head, reasons include: we don't want to optimize FP error, limited precision in the input data, and negligible real world impact.
That said, it's also well known that non-OR practitioners have less confidence in our results when there are trivial local suboptimalities, in some cases even when they don't affect the objective function (e.g., off the critical path in a scheduling problem); I've heard of several professionals who pass the output of exact (modulo stopping criteria) methods through stupid local searches just for that reason.
Concorde produces a provably optimal tour, but it follows the TSPLIB input format and requires that all distances be integers. There will thus be rounding error in converting the geodesic distances to integers. To obtain greater precision, the geodesic distances should be scaled to meters rather than kilometers.
That said, it's also well known that non-OR practitioners have less confidence in our results when there are trivial local suboptimalities, in some cases even when they don't affect the objective function (e.g., off the critical path in a scheduling problem); I've heard of several professionals who pass the output of exact (modulo stopping criteria) methods through stupid local searches just for that reason.