If there are significant differences (as measured by a professional) then there's good reason to run more trials. Actual data from well-run clinical trials always trumps anecdotal evidence though.
This actually happened in the case of vaccines again and again. The trials have repeatedly shown no link between vaccines and autism.
A long term third party independently funded study vaxxed / non vaxxed with an inert placebo would put this to bed once and for all.
As we have discussed elsewhere the only one done to this standard was the flu vaccine study which was limited to a few hundred people. Whilst us fighting for justice for vaccine damaged kid see it as important, the small sample size means that the Pharma apologists can dismiss it and say that it does not count.
Obviously if it was to be found that the risks of vaccines had been understated and/or the benefits overstated it would be the biggest medical scandal of all time. They are so many people that staked their careers and reputations on promoting more and more mass vaccination that it is difficult to see turkeys voting christmas by supporting this type of study. (see CDC refusal to release patient level data of their trials to third parties and alleged manipulation of data which is the subject of the film Vaxxed)
These are some of the major obstacles to overcome but I remain hopeful. The best bet rather frighteningly is Trump who remains very concerned about vaccine safety. He has a vaccine damaged child in his family. However, I fear that war would have destroyed the World before the trial could be completed if he was elected.