Probably worth mentioning that there is also the CSS Grid Layout Module [1], which has (according to Can I use...) much less support (9.65% [2]) vs Flexbox (79.32% [3]). A quick survey suggests that Flexbox should be used 'in the small' whilst grid is for 'in the large', but Solved by Flexbox seems to suggest that it is equally competent as a solution for whole page grid layout. Is the CSS Grid Layout module a dead end?
CSS Grid Layout is definitely not a dead end, it just didn't have enough support to showcase (yet).
As for the difference, Flexbox is for one-dimensional layout and Grids are for two-dimensional layout.
Grids will eventually be able to do what you now need a flexbox within a flexbox to do, and much more. They'll be especially helpful for layouts that need to be dramatically different across media breakpoints.
[1] http://www.w3.org/TR/2013/WD-css3-grid-layout-20130910/ [2] http://caniuse.com/#search=grid [3] http://caniuse.com/#feat=flexbox