"We didn't talk for a long time." (simple past tense) You might say this when you are describing a past situation that has ended, since you are talking with that person again. It could be that you and your best friend had a fight and refused to talk with each other. But now you both have made up and are talking to each other again. "We haven't talked for a long time." (present perfect tense) You didn't have a fight or anything like that with your best friend. Maybe you've both been really busy and that's why you haven't seen or talked to each other for awhile.

We haven't talked for a long time. It's present perfect. Something happened in the past but effect is remained! here are some examples: I haven't seen you for a long time. I have never seen this movie. I have never listened to this music.