Why do some academic journals requires a separate "summary" paragraph in addition to an abstract? At least in the field of biology, some academic journals have added a separate summary paragraph (for example: "Significance Statement" (in Journal of Neuroscience), or "Significance" (in PNAS), or "In Brief" (in Current Biology)) in recent years. Why do the publishers think an abstract is not enough?
They serve slightly different purposes.

Specifically, the abstract is what you'd tell your coworkers; the significance statement is more like something you'd tell your mom.  

The abstract is meant to quickly summarize this particular paper. A good abstract will provide a little bit of context or background, lest readers wonder why you're studying gene X or brain area Y, but the bulk of the abstract is approach and results. "Here, we show...." The significance statement is meant to put your work in a broader context and explain/justify why your article is worth publishing. (I suspect these are also helpful in attracting media attention, if that is of interest).

For example, suppose your paper investigates an antibiotic resistance gene. The abstract might look somethng like this one from MacMahon et al. (2009):


  The ardA gene, found in many prokaryotes including important
  pathogenic species, allows associated mobile genetic elements to evade
  the ubiquitous Type I DNA restriction systems and thereby assist the
  spread of resistance genes in bacterial populations. As such, ardA
  contributes to a major healthcare problem. We have solved the
  structure of the ArdA protein from the conjugative transposon Tn916
  and find that it has a novel extremely elongated curved cylindrical
  structure with defined helical grooves....


This briefly hits on the context, but doesn't really delve into it. A significance statement would talk more about this healthcare problem (e.g., X patients have multidrug resistant infections) and how this paper helps solve it (target for co-treatments or whatever?). 

Edit: take a look at the PLoS Computational Biology guidelines and examples (linked therein) that @cheersmate found for some more direct comparisons.