The Sneak perk makes you harder to detect when sneaking. At higher ranks, you no longer set off mines or other floor-based traps, running no longer affects detection, and simply entering stealth mode may make enemies lose track of you.”— Fallout 4 loading screen hints
As the name would suggest, this perk tree focuses on sneaking around better. The first rank makes sneaking 20% easier. The second rank makes sneaking 30% easier and makes it so you no longer trigger floor based traps. The third rank increases the ease of sneaking by 40% and makes it so that you no longer trigger enemy mines. The fourth rank makes sneaking 50% easier and allows one to run without adversely affecting stealth. The fifth and final rank allows you to sneak and have far away enemies lose sight of you.