Clawshot
The Clawshot is an item introduced in Twilight Princess. It is based off of the Hookshot. Instead of having just a point, it has three claws that are used to grab things to pull them closer to Link or pull Link to them.
Link first received the Clawshot in Lakebed Temple at the bottom of Lake Hylia, and it becomes important for beating that temple's boss, Morpheel.
In the City in the Sky, Link received another Clawshot, which he used in conjunction with each other. Collectively, they are called the Double Clawshot and they enable one to cling onto one thing, and Clawshot to another, flying from one side of the area to the other. These two items combined make it easy to win the second Star Game set up for you in Castle Town.
The Clawshot is Link's tether recovery in Super Smash Bros. Brawl
Double Clawshot
While Twilight Princess features a variation of the hookshot, once reaching the seventh dungeon (City in the Sky) Link'll receive the Double Clawshots, which simply is another Clawshot, one for each hand. This item works very similarly to Spider-Man's web technique, enabling Link to grapple onto an object while holding onto (or swinging from) another. This is required to go deeper into the dungeon.
The visible difference is that previously, when Link grappled onto a target, his only option was to drop to the platform below. However, the combination of two clawshots allows his to grapple and pull himself towards one target, then grapple another target without the need to drop down before hand. This is vital, since many sections of the City in the Sky have pitfalls and holes instead of solid platforms.
The Double Clawshots are the key item to defeating the boss of the city in the sky, Argorok.
Clawshot vs. Hookshot
The Clawshot is a departure from the Hookshot, since it can no longer attach to the same variety of items as previously possible. Where the Hookshot could latch on to torches, chests, rooftops, and other indicated targets, the clawshot is more limited in what objects it can grasp. These objects are typically a form of grating, which somewhat diminishes the variety of different "Hookshot-table" areas.
On the other hand, the Clawshot surpasses the hookshot in other areas. The ability to now cling to grates once you have reached the target is a nice ability, and in conjunction with the second Clawshot, opens up interesting new exploration abilities. The ability to move up and down along the Clawshot's chain is somewhat underutilized within Twilight Princess, but it's a nice ability, reminiscent of The Wind Waker's grappling hook.