Hookshot and Clawshot
The Hookshot, or the Clawshot in Super Smash Bros. Brawl and Smash 4, is a grab move performed by Link in Super Smash Bros., Link and Young Link in Super Smash Bros. Melee, and Link and Toon Link in Super Smash Bros. Brawl and Smash 4. The Hookshot is a recurring item in the Legend of Zelda universe, and the Clawshot has also featured in more than one Zelda game, albeit with a different appearance in each. Samus uses a similar move in all three games, called the Grapple Beam.
The Hookshot is Link's grab in Super Smash Bros., Link and Young Link's grabs in Super Smash Bros. Melee, and Toon Link's grab in Super Smash Bros. Brawl. In Super Smash Bros., the hook of the Hookshot fires straight out, and then returns straight back, like a spring. In Melee and Brawl, however, the Hookshot/Clawshot (respectively) fires straight out, falls to the ground, and is then slung back towards Link like a rope. Also in Melee, Link/Young Link use this to perform the wall-grapple, since the Hookshot can latch to essentially any wall at any point at appropriate distances. Using the Hookshot on an incline causes it to retract immediately, which often leaves Link vulnerable for short periods of time. This is notable on Princess Peach's Castle. Using this technique, one can use mindgames to assist in recoveries by choosing either to sweet spot the ledge or grapple another point of the wall; however, should the wall-grapple miss or otherwise fail, Link/Young Link will be left in the "helpless" animation. Additionally, the Hookshot may be used immediately after an air dodge, meaning a directional air dodge can give additional distance before using the Hookshot.
In Super Smash Bros. Brawl, Toon Link uses his to perform the tether recovery and grab. Similar to Link's Clawshot, if it misses, Toon Link is left wide open. It also has two hitboxes that follow the same mechanic from Link's Clawshot. In Brawl, Toon Link's Hookshot has a shorter range in grabs and tethers compared to Link's Clawshot, but this makes it slightly faster. As it is the case with all tether recoveries in Brawl, the Clawshot/Hookshot may only latch onto the ledge, making edgehogging the recovery relatively easy.
The Clawshot is Link's grab in Super Smash Bros. Brawl and Super Smash Bros. 4. Although it is very similar to the the Hookshot in Super Smash Bros. Melee, not only does it retract more quickly, but Link's new tether recovery now finds and tracks the wall, automatically sweetspotting a nearby ledge instead of just latching onto it when it touches. Also, when used in the air to attack, there are two different hitboxes. The first is when the claw is shot out, and hits the opponent. The second hitbox is at the end of the Clawshot, when the claw closes. Also, if Link lands while using the Clawshot in the air, the move auto-cancels, which is a useful technique when combined with SHFF, quite possibly Link's best tactic for interrupting other characters' approaches. If used in contact with the ground, the move becomes a grab, and like all other tether grabs is laggy if missed. It should also be noted that if it fails, Link doesn't enter a Helpless state. In most cases, it's recommended to use the Clawshot for recovery because it helps prevent Link from getting edgeguarded by certain attacks.
The Clawshot has an invaluable edgehogging property in that it can quickly give Link more invincibility frames by pressing back after Link has grabbed the edge, and immediately pressing "Z" releasing the Clawshot, and pressing "Z" again to instantly grab the ledge and gain invincibility frames. This technique is sometimes referred to as Zair Edgeguarding, even though it is used as an edgehogging technique.
The reason that the two are named differently is that Link in Super Smash Bros. Brawl is based off of his Twilight Princess model, where the item in the game is called the "Clawshot" instead of the old "Hookshot" from Ocarina of Time.