I performed two RNA-Seq experiments in exactly the same way, with the only difference being the studied cell type (mouse cortical neurons vs sensory DRG neurons). Everything about these experiments suggests they went really well, and the enriched genesets make a lot of sense.

About the attached volcano plots: Do I need to be concerned about the fact that the p value range is smaller in one experiment ? If the reason is in the underlying biology (and not a technical issue), is the explanation that a higher variability of individual measurements is sometimes observed ?

The adjusted p-values are really just for thresholding at some value and giving you a bounded set.

Whether it's 1e-10 or 1e-100, it's just telling you how, under the null model, it would be unlikely to see such strong separation of two groups with such low within group variance. But I don't typically compare smallest p-values from one experiment vs another, I don't know what that would tell you.