At the very high level:
If you have weights from a survey data set they might be doing quite a few things, the most straightforward of which is allowing you to offset the survey's sampling scheme. For example if women were over-sampled relative to men then the weights would reflect this and analyses that used them would be correct for the actual population's gender balance rather than the one in the data. In your case, they might be offsetting the standardization too.
In short, weights change your estimand (the quantity your estimation strategy is targeting). So if you care about the quantities your survey thinks you ought to care about, e.g. to be 'representative' to a particular population, then you'd want to use its weights.
But things are, inevitably, more complicated than that, as weights can offset other features and perhaps less necessary when your model's covariates include the one used to unbalance the sample, or when you want particular conditional effects.
The best advice is to take a look at the survey's variable codebook and see what it thinks the weights will do for you. (There may indeed be different weights for different purposes). Then make your decisions on that basis. Certainly not on whether the model summaries look different with and without them.
