Speaks to usually means a conversation. Speaks with can mean the same, but it is often used to mean to speak to someone a particular person about a particular subject, or complaint, or annoyance etc. If you are in a school and the teachers says "The school governor wants to speak with you", it usually means bad news or he has something to tell you. "the school governor wants to speak to you" would imply only a chat, but this use is dying out in the UK you might still come across it. Apart from that they are the same