To me, She's different from me implies a stronger sense of difference, while She's not like me more simply points out a lack of similarity. At the extremes (#1 and #2 below) we can have:
1. She's different from me. I couldn't tell a lie to save my life, and she doesn't know what 'truth' is.
2. She's not like me. I simply have to tell the truth as I see it, but she will tell the odd white lie to avoid hurting someone's feelings.
In #3 and #4, the only difference is the speaker's feeling. In #3, she feels the difference more strongly than in #4:
3. She's different from me. She likes sport, while I prefer more leisurely activities.
4. She's not like me. She likes sport, while I prefer more leisurely activities.
However, I have a sneaking suspicion that I have tried to convince myself that there is a difference when, in reality, there isn't.