There is hardly any difference between talk to and talk with. I think the form talk with is more common in American English.
- Can I talk with Alice, please? OR Can I talk to Alice, please?
- I want to talk to you. OR I want to talk with you.
- You must talk to your boss. OR You must talk with your boss.
To talk to / talk with somebody is to have a meaningful conversation with them. This is usually a two-way process. That means both parties exchange their ideas.
When you talk at somebody, you are just talking at them. You don’t care whether they are listening or not. You are also not keen on hearing what they have to say. Note that this expression has a negative connotation.
- He was a difficult boss. He would always talk at you. He wasn’t really interested in hearing what you had to say.
- Don’t just talk at me. I also want to be heard.
- The teacher was merely talking at her students.