In Pandemic, why take the extra step of eradicating a disease after you've cured it?
Eradicating a disease relieves pressure on the game board. Those infection cards that correspond to the eradicated disease remain in the deck, and continue to get drawn but have no effect other than counting as a drawn card.

If a quarter of your infection deck is red and you are drawing 3 infection cards each turn, and you eradicate red, then a quarter of the infection cards have no effect, and your effective is 3/4*3 = 2.25 cards per turn. 

If the top of the infection deck is weighted to a particular colour, the change can be more noticeable. If 2/5 cards are of the same colour and you eradicate it, your effective draw rate at 3 cards per turn will be 1.8 cards per turn.

The less infections on the board, the easier your team will have managing the rest of the infections. The game has a limited number of total actions possible (by the number of cards in the player deck) by the players, and removing a colour from the board frees actions that would be spent controlling the disease and allows more actions spent on other colours. It's a fine balance between how many more actions it takes to eradicate a disease and what that frees up in the future. 

As James mentioned in a comment below, there's another factor for eradicating a disease:


  You also care if you run out of blocks of a particular color and are required to place another one, because that is a game loss.


And as gambler22 says, you get a potential relief on the Infect step of the Epidemic cards:


  You don't only get potential relief at the top of the deck, but at the bottom of the deck as well if an Epidemic card is drawn.