Nothing.
Well at least nothing if you're measuring their weights, body fat percentages, appetites, blood sugars, insulins, incretins (GLP1), lipids, fasting leptin levels, or liver function tests. Those were in fact all the things that were looked at in this recent study.
Sorry chemophobes.
(but conspiracists can still rest easy knowing that funding was provided in part by aspartame producer Ajinomoto, and of course it would be nice to see this study repeated with study participants with obesity)
(And if you're interested here's a recent review that looked at both observational studies on non-nutritive sweeteners and randomized trials looking at non-nutritive sweeteners which found that while observational studies suggest risk to weight, to date, randomized trials, now including this one it would seem, don't).