"To speak" to someone and "to talk" to someone are basically the same expression, but there's a couple differences between the two verbs. Generally, "to speak" is used in more formal situations and often for more polite requests, e.g. "I would like to speak with him." It's also more for the act of actually saying something, while "talk" is engaging communication with others. This might sound confusing, so I'll give some examples: "speak up," "speak out," "actions speak louder than words" In all the situations above, "talk" would be inappropriate. The actions are figuratively saying something and not actually engaging in communication (if that makes sense). Another example: I would like to speak to the king. I would like to talk to the king. The second one with "talk to" appears more conversational than the first one. The main difference between these two words is that "talk" implies more of a conversation, while "speak" is more one-sided. Hopefully I explained that clearly!

In many situations either word can be used. As Leila said, sometimes "speak" is a little more formal.