What is the difference?

The first one means you were talking to her and then you stopped. This could be one instance..for example, I stopped talking to her because I had to catch a bus, or more long term.....My friend kept upsetting me whenever I phoned so I stopped talking to her for a week

The second one is a specific action...I saw my friend on the street and I stopped to talk to her. In this sentence stopped means you stopped yourself (from walking. driving...) In the first example it was the talking that stopped.

What is the difference?

I stopped talking to her- You are not talking to her anymore. Usually I hear people use this when they are not on good terms with the person. The example that Sarah gave is good.

I stopped to talk to her- You are stopping what you are doing to talk to her. Example- I was walking down the street when I saw her, so I stopped to talk to her.