"Speak to" vs. "Speak with"

Solution 1:

These two are more or less equivalent. They can both be used for the situation where A and B speak to each other. "Speak to" can also be used for the situation where A talks and B listens without speaking.

Solution 2:

You are more likely to encounter speak with in American English, which employs the verb + with construction (speak with, meet with) very much more than British English does.

I remember hearing a clip of Laura Bush speaking on the radio (I think to the people of Afghanistan) and saying how pleased she was to be "speaking with you" and to British English ears it sounded very odd. On the radio, how are you doing anything other than speaking to your audience?

Solution 3:

I have seen "Speak to" being used when there is a monologue kind of a situation, A speaks and B listens. Refer Shinto's answer. This has reference to instructing, reprimanding and situations like this.

"Speak with" is a more neutral kind of a term implying a bidirectional communication.