The SR5
Beginner Box is mainly for younger, beginning role-players. It has some ready-to-play character sheets and a copy of the Quick Start Rules (which you can download for free from the Catalyst website,
http://www.shadowruntabletop.com/game-resources/). I don't care for the QSR, because they oversimplify and even ignore some key mechanics (like drain), which can cause confusion later.
The SR5
Alphaware Box is much better. There are expanded versions of the character sheets from the Beginner Box. There's a set of spell cards and gear cards, which can be neat props at the gaming table. There's a poster map of North America. There's a "Rules of the Street" booklet that includes an abridged set of rules that makes a handy table reference. It has gaps, but (unlike the QSR) at least it doesn't directly contradict the core rules, so it remains relevant and useful. And there's also an excellent mini-campaign booklet called "Plots and Paydata," which outlines a series of six shadowruns designed for beginning players and GMs. Highly recommended.