Sometimes (not always!) по- adds the sense of 'doing something for a while - почитать 'to read for a while,' поспать 'to take a nap.' I think it's a similar sense here, as Dmitry points out - have a conversation, chat for a bit, as opposed to just saying something.

I guess "поговорить" can be translated like "to speak" in the sense of "to have a conversation". "Я хочу поговорить с тобой" - "I want to speak to you [to have a conversation with you]". That means the conversation will take some time.

The answers above are correct. In practice the difference is the following: Я говорил с ним (instrumental)- I was speaking to him. Probably several times (imperfective verb feature) Я поговорил с ним (instrumental) - I spoke to him. And I have some results, probably in the next sentence I'll tell you about them. Я сказал ему(dative), что у меня нет времени - I told him ("с ним" is not possible here) that I didn't have time.

Both answers below are correct. Поговорить means 'to talk with someone for a certain amount of time'. It's also close to the verb 'to discuss' = to talk with someone about something and tell each other your ideas or opinions.