They are the same thing right? I mean they both prevent you to move while you can still cast some non movement abilities and auto-attacks.
League of Legends technically has no CC by the name of Snare. While some tooltips may say that they snare a target (though I cannot recall the flavor text of every ability in the game and cannot be sure), the two types of CC that people would usually mean when they say Snare are Slow and Root.
There has always been a bit of ambiguity in the terms as used by players though due to cross-over with other games. This mainly comes from World of Warcraft players, as in WoW, a slow is called a snare. If someone uses the terminology in League, make sure to get clarification on if they mean a slow or a root if you are not sure.
A list of CCs in the game can be found at this link.
There's no difference in functionality between the 2. The difference is only in flavour text.
Snare: a trap for catching birds or animals, typically one having a noose of wire or cord. Example: Yordle Snap Trap, Light Binding Root: establish deeply and firmly. Example: Curse of the Sad Mummy, Nevermove