What is the difference/benifit of golden cards?

Hearthstone - Golden Cards

Golden cards are special, rarer versions of cards. Each card exists in both a regular and a golden form. The difference between a card's normal version and its golden version is purely aesthetic - all golden cards feature a golden frame, and each has a unique animation.

There is no difference in gameplay or behaviour on the battlefield, and golden cards do not allow you to circumvent the 2-per-deck (1-per-deck for legendaries) card restriction.

Golden cards are commonly compared with "foil" cards of other Trading Card Games, and are far harder and more costly to obtain than their regular versions. Golden cards are intended to allow players to show off their card-collecting achievements.

While they may contain no tactical advantage than regular cards, the main and most important difference is that they will grant more dust during disenchanting; which varies by rarity

Hearthstone - Crafting

As @Sconibulus mentioned, you have a chance to obtaining a Gold Coin if your deck consists entirely of gold cards.

Hearthstone - The Coin

If a player's deck is composed entirely of golden cards, they will be granted a golden version of The Coin when going second.

Another cool aspect, which is purely cosmetic, is that if you play a gold card that replaces or upgrades your Hero Power, you will obtain a gold hero power

Replacing Hero Powers (cited from Gold Card link)

• Cards which replace a Hero Power with another one (such as Lord Jaraxxus, Sir Finley Mrrgglton or Shadowform) will only grant golden Hero Powers if the card is golden. For example, playing a regular Sir Finley Mrrgglton with a golden Hero Power will replace it with a regular Hero Power; playing a regular Shadowform after playing a golden Shadowform will replace the golden Mind Spike with a regular Mind Shatter.
• The one exception is Justicar Trueheart, which will produce a golden Hero Power if either the card or the Hero Power is golden. This may be because the card conceptually "upgrades" the Hero Power.

In terms of game mechanics they give more dust when you convert them, double that of their non-gold counterparts.

Other than that, they're just there to look cool. (the pictures also have slight movement)

Golden Cards have several benefits, none of them tactile save two.

• All cards may be dis-enchanted for dust, this dust can be used to 'craft' specific cards you want, therefore circumventing the 'random' nature of the packs you open. Golden Cards grant twice as much dust as their normal counterparts. This is the first tactile benefit.

• Golden cards count as an additional card not an upgrade to an existing one, so if you have one Normal Card and one Golden Card you can put both in your deck to meet the 2 card maximum. This is the second tactile benefit.

• Golden Cards have the visual benefit of having animated pictures that often are more interesting to look at than normal ones and flaunt some flair for you as you play. This obviously has no real impact on the game itself.

• Golden Cards that have abilities that create other cards will share their golden animated picture benefit to their produced offspring. Once again this is just for show.

• If a player's deck is entirely composed of Golden Cards and they start the game going second, they will receive a Golden Coin instead of a Normal Coin. Thanks @Sconibulus for this contribution!