In response to Susan's comment:

 The one instance where we always use 'speak' rather than 'talk' is with reference to language ability. 'Do you speak English?' 'He learnt to speak French' 'He couldn't speak Arabic.' We cannot use 'talk' in these cases.

'Speak' can be used to mean simply 'address',  while 'talk' suggests chatting, with meaningful content of some kind. For example, if somebody says that they were at a party where there was a famous person present, you might ask 'Did he speak to you?'. This might just mean 'Hello' or 'Thanks' or 'Where's the bathroom?'. But if you say 'Did he talk to you?', this would mean an actual conversation on a more personal level, with some kind of exchange of information or ideas.

 There are also set phrases in which we use one rather than the other, for example  'Speak up' or 'talk it over' and so on.

 Otherwise, as Chirag says, the difference is a subtle level of formality. 'Speak' is more formal, and can sometimes mean 'give a speech'. For example, 'He spoke at the annual ceremony', suggests giving a formal address. 'The President came out onto the the balcony and spoke to the crowd of people.' This suggests a public speech. 'Talk' would suggest a level of personal intimacy that wouldn't be appropriate. 

In most cases, though, they are virtually interchangeable.

The word speak is used when you are related to a person.

Jhoane spoke to Anthony yesterday.

The word talk is used to describe nouns or discussing events.

Jhoane will talk on the issue of Global Warming in the next session.