Is there a difference or is riot just bad with vocabulary?
Bad with vocabulary.
Roots, snares, and immobilizes all drop movement speed down to 0 and silence pretty much every skill that requires moving the champion to do something (a few mobility skills like Caitlyn's 90 Calibure Net can still be cast) but it's just safe to assume you can't use anything with mobility. Fiora can't cast Blade Waltz, Riven can't cast Valor - generally such skills are just flat out disabled.
I've read something from a Rioter where Riot decided to change the name of snares to roots somewhere along the lines (after a Rioter actually looked up what snares in RL do). For clarity - though they haven't updated all the tooltips yet.
As for potential differences - Amumu's ultimate (an "entangle") is the only root I'm aware of that functions differently than simply preventing mobility, as it also disables auto-attacks. It used to also silence champions, so for a while it basically read "Rito, you're high - why are you pretending a stun does something besides stun things?" but they changed it to actually do what the tooltip says somewhere along the line, so now it's essentially a root with a fairly unique AA disabling bonus.