I'm sure there's a lot to say about the relationship between Whitehead's "Science and the Modern World" and its relationship with Radical Empiricism. Here's what occurs to me—I hope it works as a place to start!
Whitehead was preoccupied with the worry that contemporary science with its materialistic emphases was moving in the wrong direction; his peers tended to believe that "only such things as could be localized at a mathematically simple 'point' of space and time were genuinely real" (IEP). Thus "relations and connections" were disregarded in favor of the "real" and physically locatable. Whitehead believed this to be a grave mistake—fields of relations should be prioritized even above the physical, as he believed they were in nature. The resulting fallacy led physicists to regard the abstract as concrete.
William James's conception of Radical Empiricism seems (to me) to support Whitehead's concerns and combat them by postulating (among other things) that both experience and relations between experiences are valid parts of scientific explanations. Scientific explanations should not confine themselves to the physical level (mere "whirling particles") but should also consider the bigger picture: the connections that explain and give meaning. Relations between things are what provides those things with meaning.