First off, you have fig envy. It is a known and well documented disease. Not curable.
Second, check your neighbor's tree to see if the figs are forming on OLD wood, or are forming on the NEW growth. if they are on the old growth (from last season) they are breba figs. It may be that your neighbor has a variety that forms breba figs, and your variety does not. Nothing wrong with that, and no way to fix it, apart from getting a different tree.